Theorem opsqrlem6 31948

Description: Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
opsqrlem2.1	⊢ 𝑇 ∈ HrmOp
opsqrlem2.2	⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))
opsqrlem2.3	⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
opsqrlem6.4	⊢ 𝑇 ≤op Iop

Assertion

Ref	Expression
opsqrlem6	⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

Distinct variable group:   𝑥,𝑦,𝑇

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem opsqrlem6

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6891	. . 3 ⊢ (𝑗 = 1 → (𝐹‘𝑗) = (𝐹‘1))
2	1	breq1d 5152	. 2 ⊢ (𝑗 = 1 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3		fveq2 6891	. . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
4	3	breq1d 5152	. 2 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5		fveq2 6891	. . 3 ⊢ (𝑗 = 𝑁 → (𝐹‘𝑗) = (𝐹‘𝑁))
6	5	breq1d 5152	. 2 ⊢ (𝑗 = 𝑁 → ((𝐹‘𝑗) ≤op Iop ↔ (𝐹‘𝑁) ≤op Iop ))
7		opsqrlem2.1	. . . 4 ⊢ 𝑇 ∈ HrmOp
8		opsqrlem2.2	. . . 4 ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))
9		opsqrlem2.3	. . . 4 ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
10	7, 8, 9	opsqrlem2 31944	. . 3 ⊢ (𝐹‘1) = 0hop
11		idleop 31934	. . 3 ⊢ 0hop ≤op Iop
12	10, 11	eqbrtri 5163	. 2 ⊢ (𝐹‘1) ≤op Iop
13		idhmop 31785	. . . . . . . 8 ⊢ Iop ∈ HrmOp
14	7, 8, 9	opsqrlem4 31946	. . . . . . . . 9 ⊢ 𝐹:ℕ⟶HrmOp
15	14	ffvelcdmi 7087	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ HrmOp)
16		hmopd 31825	. . . . . . . 8 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘𝑘) ∈ HrmOp) → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
17	13, 15, 16	sylancr 586	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)) ∈ HrmOp)
18		eqid 2727	. . . . . . . 8 ⊢ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))
19		hmopco 31826	. . . . . . . 8 ⊢ ((( Iop −op (𝐹‘𝑘)) ∈ HrmOp ∧ ( Iop −op (𝐹‘𝑘)) ∈ HrmOp ∧ (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
20	18, 19	mp3an3 1447	. . . . . . 7 ⊢ ((( Iop −op (𝐹‘𝑘)) ∈ HrmOp ∧ ( Iop −op (𝐹‘𝑘)) ∈ HrmOp) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
21	17, 17, 20	syl2anc 583	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp)
22		leopsq 31932	. . . . . . 7 ⊢ (( Iop −op (𝐹‘𝑘)) ∈ HrmOp → 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))))
23	17, 22	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))))
24		opsqrlem6.4	. . . . . . . 8 ⊢ 𝑇 ≤op Iop
25		leop3 31928	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇)))
26	7, 13, 25	mp2an 691	. . . . . . . 8 ⊢ (𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇))
27	24, 26	mpbi 229	. . . . . . 7 ⊢ 0hop ≤op ( Iop −op 𝑇)
28		hmopd 31825	. . . . . . . . 9 ⊢ (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iop −op 𝑇) ∈ HrmOp)
29	13, 7, 28	mp2an 691	. . . . . . . 8 ⊢ ( Iop −op 𝑇) ∈ HrmOp
30		leopadd 31935	. . . . . . . 8 ⊢ ((((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp ∧ ( Iop −op 𝑇) ∈ HrmOp) ∧ ( 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∧ 0hop ≤op ( Iop −op 𝑇))) → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
31	29, 30	mpanl2 700	. . . . . . 7 ⊢ (((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp ∧ ( 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∧ 0hop ≤op ( Iop −op 𝑇))) → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
32	27, 31	mpanr2 703	. . . . . 6 ⊢ (((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) ∈ HrmOp ∧ 0hop ≤op (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘)))) → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
33	21, 23, 32	syl2anc 583	. . . . 5 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)))
34		2cn 12311	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
35		hmopf 31677	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘): ℋ⟶ ℋ)
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘): ℋ⟶ ℋ)
37		homulcl 31562	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
38	34, 36, 37	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ)
39		hmopf 31677	. . . . . . . . . . 11 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
40	7, 39	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
41		fco 6741	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
42	36, 36, 41	syl2anc 583	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)
43		hosubcl 31576	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
44	40, 42, 43	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
45		hmopf 31677	. . . . . . . . . . . 12 ⊢ ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
46	13, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ Iop : ℋ⟶ ℋ
47		homulcl 31562	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
48	34, 46, 47	mp2an 691	. . . . . . . . . 10 ⊢ (2 ·op Iop ): ℋ⟶ ℋ
49		hosubsub4 31621	. . . . . . . . . 10 ⊢ (((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
50	48, 49	mp3an1 1445	. . . . . . . . 9 ⊢ (((2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
51	38, 44, 50	syl2anc 583	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
52		hosubcl 31576	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ)
53	42, 38, 52	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ)
54		hoadd32 31586	. . . . . . . . . . . . . . 15 ⊢ (( Iop : ℋ⟶ ℋ ∧ (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ) → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
55	46, 46, 54	mp3an13 1449	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
56	53, 55	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
57		ho2times 31622	. . . . . . . . . . . . . . 15 ⊢ ( Iop : ℋ⟶ ℋ → (2 ·op Iop ) = ( Iop +op Iop ))
58	46, 57	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2 ·op Iop ) = ( Iop +op Iop )
59	58	oveq1i 7424	. . . . . . . . . . . . 13 ⊢ ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) = (( Iop +op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))))
60	56, 59	eqtr4di 2785	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
61		hoaddsubass 31618	. . . . . . . . . . . . . 14 ⊢ (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
62	48, 61	mp3an1 1445	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
63	42, 38, 62	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ((2 ·op Iop ) +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
64	60, 63	eqtr4d 2770	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))))
65	64	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇))
66		hoaddcl 31561	. . . . . . . . . . . 12 ⊢ (( Iop : ℋ⟶ ℋ ∧ (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘))): ℋ⟶ ℋ) → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
67	46, 53, 66	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ)
68		hoaddsubass 31618	. . . . . . . . . . . 12 ⊢ ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
69	46, 40, 68	mp3an23 1450	. . . . . . . . . . 11 ⊢ (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))): ℋ⟶ ℋ → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
70	67, 69	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
71		hoaddcl 31561	. . . . . . . . . . . 12 ⊢ (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
72	48, 42, 71	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ)
73		hosubsub4 31621	. . . . . . . . . . . 12 ⊢ ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
74	40, 73	mp3an3 1447	. . . . . . . . . . 11 ⊢ ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
75	72, 38, 74	syl2anc 583	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
76	65, 70, 75	3eqtr3d 2775	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
77		hosubadd4 31617	. . . . . . . . . . . 12 ⊢ ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
78	40, 77	mpanr1 702	. . . . . . . . . . 11 ⊢ ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
79	48, 78	mpanl1 699	. . . . . . . . . 10 ⊢ (((2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
80	38, 42, 79	syl2anc 583	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (((2 ·op Iop ) +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((2 ·op (𝐹‘𝑘)) +op 𝑇)))
81	76, 80	eqtr4d 2770	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = (((2 ·op Iop ) −op (2 ·op (𝐹‘𝑘))) −op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
82		halfcn 12451	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
83		homulcl 31562	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℂ ∧ (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
84	82, 44, 83	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ)
85		hoadddi 31606	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ (𝐹‘𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ) → (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·op (𝐹‘𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
86	34, 85	mp3an1 1445	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ) → (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·op (𝐹‘𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
87	36, 84, 86	syl2anc 583	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·op (𝐹‘𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
88		2ne0 12340	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
89	34, 88	recidi 11969	. . . . . . . . . . . . 13 ⊢ (2 · (1 / 2)) = 1
90	89	oveq1i 7424	. . . . . . . . . . . 12 ⊢ ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
91		homulass 31605	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ) → ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
92	34, 82, 91	mp3an12 1448	. . . . . . . . . . . . 13 ⊢ ((𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ → ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
93	44, 92	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((2 · (1 / 2)) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
94		homullid 31603	. . . . . . . . . . . . 13 ⊢ ((𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))): ℋ⟶ ℋ → (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
95	44, 94	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
96	90, 93, 95	3eqtr3a 2791	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))) = (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
97	96	oveq2d 7430	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((2 ·op (𝐹‘𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
98	87, 97	eqtrd 2767	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))) = ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
99	98	oveq2d 7430	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op ((2 ·op (𝐹‘𝑘)) +op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
100	51, 81, 99	3eqtr4d 2777	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
101		hoaddcl 31561	. . . . . . . . 9 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))): ℋ⟶ ℋ) → ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
102	36, 84, 101	syl2anc 583	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ)
103		hosubdi 31611	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ ∧ ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ) → (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
104	34, 46, 103	mp3an12 1448	. . . . . . . 8 ⊢ (((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))): ℋ⟶ ℋ → (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
105	102, 104	syl 17	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
106	100, 105	eqtr4d 2770	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)) = (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
107		hosubcl 31576	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
108	46, 36, 107	sylancr 586	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ)
109		hocsubdir 31588	. . . . . . . . . 10 ⊢ (( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ ∧ ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))))
110	46, 109	mp3an1 1445	. . . . . . . . 9 ⊢ (((𝐹‘𝑘): ℋ⟶ ℋ ∧ ( Iop −op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))))
111	36, 108, 110	syl2anc 583	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))))
112		hmoplin 31745	. . . . . . . . . . . . . . 15 ⊢ ( Iop ∈ HrmOp → Iop ∈ LinOp)
113	13, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Iop ∈ LinOp
114		hoddi 31793	. . . . . . . . . . . . . 14 ⊢ (( Iop ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
115	113, 46, 114	mp3an12 1448	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
116	36, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))))
117	46	hoid1i 31592	. . . . . . . . . . . . . 14 ⊢ ( Iop ∘ Iop ) = Iop
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119		hoico2 31560	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ( Iop ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
120	36, 119	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ (𝐹‘𝑘)) = (𝐹‘𝑘))
121	118, 120	oveq12d 7432	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
122	116, 121	eqtrd 2767	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop −op (𝐹‘𝑘)))
123		hmoplin 31745	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ HrmOp → (𝐹‘𝑘) ∈ LinOp)
124	15, 123	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ LinOp)
125		hoddi 31793	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
126	46, 125	mp3an2 1446	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ LinOp ∧ (𝐹‘𝑘): ℋ⟶ ℋ) → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
127	124, 36, 126	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
128		hoico1 31559	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → ((𝐹‘𝑘) ∘ Iop ) = (𝐹‘𝑘))
129	36, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ Iop ) = (𝐹‘𝑘))
130	129	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (((𝐹‘𝑘) ∘ Iop ) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
131	127, 130	eqtrd 2767	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘))) = ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))
132	122, 131	oveq12d 7432	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))) = (( Iop −op (𝐹‘𝑘)) −op ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))
133	36, 46	jctil 519	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ))
134		hosubadd4 31617	. . . . . . . . . . 11 ⊢ ((( Iop : ℋ⟶ ℋ ∧ (𝐹‘𝑘): ℋ⟶ ℋ) ∧ ((𝐹‘𝑘): ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ)) → (( Iop −op (𝐹‘𝑘)) −op ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
135	133, 36, 42, 134	syl12anc 836	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) −op ((𝐹‘𝑘) −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
136	132, 135	eqtrd 2767	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
137		ho2times 31622	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘): ℋ⟶ ℋ → (2 ·op (𝐹‘𝑘)) = ((𝐹‘𝑘) +op (𝐹‘𝑘)))
138	36, 137	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2 ·op (𝐹‘𝑘)) = ((𝐹‘𝑘) +op (𝐹‘𝑘)))
139	138	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op ((𝐹‘𝑘) +op (𝐹‘𝑘))))
140		hoaddsubass 31618	. . . . . . . . . . 11 ⊢ (( Iop : ℋ⟶ ℋ ∧ ((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
141	46, 140	mp3an1 1445	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑘) ∘ (𝐹‘𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹‘𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
142	42, 38, 141	syl2anc 583	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (( Iop +op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))) −op (2 ·op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
143	136, 139, 142	3eqtr2d 2773	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (( Iop ∘ ( Iop −op (𝐹‘𝑘))) −op ((𝐹‘𝑘) ∘ ( Iop −op (𝐹‘𝑘)))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
144	111, 143	eqtrd 2767	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) = ( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))))
145	144	oveq1d 7429	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)) = (( Iop +op (((𝐹‘𝑘) ∘ (𝐹‘𝑘)) −op (2 ·op (𝐹‘𝑘)))) +op ( Iop −op 𝑇)))
146	7, 8, 9	opsqrlem5 31947	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))
147	146	oveq2d 7430	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) = ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘)))))))
148	147	oveq2d 7430	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))) = (2 ·op ( Iop −op ((𝐹‘𝑘) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑘) ∘ (𝐹‘𝑘))))))))
149	106, 145, 148	3eqtr4d 2777	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((( Iop −op (𝐹‘𝑘)) ∘ ( Iop −op (𝐹‘𝑘))) +op ( Iop −op 𝑇)) = (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
150	33, 149	breqtrd 5168	. . . 4 ⊢ (𝑘 ∈ ℕ → 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1)))))
151		peano2nn 12248	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
152	14	ffvelcdmi 7087	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154		hmopd 31825	. . . . . 6 ⊢ (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
155	13, 153, 154	sylancr 586	. . . . 5 ⊢ (𝑘 ∈ ℕ → ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156		2re 12310	. . . . . 6 ⊢ 2 ∈ ℝ
157		2pos 12339	. . . . . 6 ⊢ 0 < 2
158		leopmul 31937	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ ( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp ∧ 0 < 2) → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
159	156, 157, 158	mp3an13 1449	. . . . 5 ⊢ (( Iop −op (𝐹‘(𝑘 + 1))) ∈ HrmOp → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
160	155, 159	syl 17	. . . 4 ⊢ (𝑘 ∈ ℕ → ( 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))) ↔ 0hop ≤op (2 ·op ( Iop −op (𝐹‘(𝑘 + 1))))))
161	150, 160	mpbird 257	. . 3 ⊢ (𝑘 ∈ ℕ → 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1))))
162		leop3 31928	. . . 4 ⊢ (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
163	153, 13, 162	sylancl 585	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hop ≤op ( Iop −op (𝐹‘(𝑘 + 1)))))
164	161, 163	mpbird 257	. 2 ⊢ (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
165	2, 4, 6, 12, 164	nn1suc 12258	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {csn 4624   class class class wbr 5142   × cxp 5670   ∘ ccom 5676  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  ℂcc 11130  ℝcr 11131  0cc0 11132  1c1 11133   + caddc 11135   · cmul 11137   < clt 11272   / cdiv 11895  ℕcn 12236  2c2 12291  seqcseq 13992   ℋchba 30722   +_op chos 30741   ·_op chot 30742   −_op chod 30743   0_hop ch0o 30746   I_op chio 30747  LinOpclo 30750  HrmOpcho 30753   ≤_op cleo 30761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cc 10452  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210  ax-addf 11211  ax-mulf 11212  ax-hilex 30802  ax-hfvadd 30803  ax-hvcom 30804  ax-hvass 30805  ax-hv0cl 30806  ax-hvaddid 30807  ax-hfvmul 30808  ax-hvmulid 30809  ax-hvmulass 30810  ax-hvdistr1 30811  ax-hvdistr2 30812  ax-hvmul0 30813  ax-hfi 30882  ax-his1 30885  ax-his2 30886  ax-his3 30887  ax-his4 30888  ax-hcompl 31005

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9380  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9527  df-card 9956  df-acn 9959  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12497  df-z 12583  df-dec 12702  df-uz 12847  df-q 12957  df-rp 13001  df-xneg 13118  df-xadd 13119  df-xmul 13120  df-ioo 13354  df-ico 13356  df-icc 13357  df-fz 13511  df-fzo 13654  df-fl 13783  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15458  df-rlim 15459  df-sum 15659  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-mulr 17240  df-starv 17241  df-sca 17242  df-vsca 17243  df-ip 17244  df-tset 17245  df-ple 17246  df-ds 17248  df-unif 17249  df-hom 17250  df-cco 17251  df-rest 17397  df-topn 17398  df-0g 17416  df-gsum 17417  df-topgen 17418  df-pt 17419  df-prds 17422  df-xrs 17477  df-qtop 17482  df-imas 17483  df-xps 17485  df-mre 17559  df-mrc 17560  df-acs 17562  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-submnd 18734  df-mulg 19017  df-cntz 19261  df-cmn 19730  df-psmet 21264  df-xmet 21265  df-met 21266  df-bl 21267  df-mopn 21268  df-fbas 21269  df-fg 21270  df-cnfld 21273  df-top 22789  df-topon 22806  df-topsp 22828  df-bases 22842  df-cld 22916  df-ntr 22917  df-cls 22918  df-nei 22995  df-cn 23124  df-cnp 23125  df-lm 23126  df-haus 23212  df-tx 23459  df-hmeo 23652  df-fil 23743  df-fm 23835  df-flim 23836  df-flf 23837  df-xms 24219  df-ms 24220  df-tms 24221  df-cfil 25176  df-cau 25177  df-cmet 25178  df-grpo 30296  df-gid 30297  df-ginv 30298  df-gdiv 30299  df-ablo 30348  df-vc 30362  df-nv 30395  df-va 30398  df-ba 30399  df-sm 30400  df-0v 30401  df-vs 30402  df-nmcv 30403  df-ims 30404  df-dip 30504  df-ssp 30525  df-ph 30616  df-cbn 30666  df-hnorm 30771  df-hba 30772  df-hvsub 30774  df-hlim 30775  df-hcau 30776  df-sh 31010  df-ch 31024  df-oc 31055  df-ch0 31056  df-shs 31111  df-pjh 31198  df-hosum 31533  df-homul 31534  df-hodif 31535  df-h0op 31551  df-iop 31552  df-lnop 31644  df-hmop 31647  df-leop 31655

This theorem is referenced by: (None)