Theorem homulass 31606

Description: Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
homulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Proof of Theorem homulass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 11217	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
2		homval 31545	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
3	1, 2	syl3an1 1161	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
4	3	3expia 1119	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
5	4	3impa 1108	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
6	5	imp 406	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
7		homval 31545	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
8	7	oveq2d 7431	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
9	8	3expa 1116	. . . . . . 7 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
10	9	3adantl1 1164	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
11		ffvelcdm 7086	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
12		ax-hvmulass 30811	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
13	11, 12	syl3an3 1163	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
14	13	3expa 1116	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
15	14	exp43 436	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥)))))))
16	15	3imp1 1345	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
17	10, 16	eqtr4d 2771	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
18	6, 17	eqtr4d 2771	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
19		homulcl 31563	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·op 𝑇): ℋ⟶ ℋ)
20		homval 31545	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
21	19, 20	syl3an2 1162	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
22	21	3expia 1119	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥))))
23	22	3impb 1113	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥))))
24	23	imp 406	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
25	18, 24	eqtr4d 2771	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
26	25	ralrimiva 3142	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
27		homulcl 31563	. . . 4 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
28	1, 27	stoic3 1771	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
29		homulcl 31563	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
30	19, 29	sylan2 592	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
31	30	3impb 1113	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
32		hoeq 31564	. . 3 ⊢ ((((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) ↔ ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇))))
33	28, 31, 32	syl2anc 583	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) ↔ ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇))))
34	26, 33	mpbid 231	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ⟶wf 6539  ‘cfv 6543  (class class class)co 7415  ℂcc 11131   · cmul 11138   ℋchba 30723   ·_ℎ csm 30725   ·_op chot 30743

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 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-mulcl 11195  ax-hilex 30803  ax-hfvmul 30809  ax-hvmulass 30811

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-map 8841  df-homul 31535

This theorem is referenced by:  homul12  31609  honegneg  31610  leopmul  31938  nmopleid  31943  opsqrlem1  31944  opsqrlem6  31949