Theorem cygznlem3 21490

Description: A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛ℤ. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
cygzn.b	⊢ 𝐵 = (Base‘𝐺)
cygzn.n	⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
cygzn.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
cygzn.m	⊢ · = (.g‘𝐺)
cygzn.l	⊢ 𝐿 = (ℤRHom‘𝑌)
cygzn.e	⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
cygzn.g	⊢ (𝜑 → 𝐺 ∈ CycGrp)
cygzn.x	⊢ (𝜑 → 𝑋 ∈ 𝐸)
cygzn.f	⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)

Assertion

Ref	Expression
cygznlem3	⊢ (𝜑 → 𝐺 ≃𝑔 𝑌)

Distinct variable groups:   𝑚,𝑛,𝑥,𝐵   𝑚,𝐺,𝑛,𝑥   · ,𝑚,𝑛,𝑥   𝑚,𝑌,𝑛,𝑥   𝑚,𝐿,𝑛,𝑥   𝑥,𝑁   𝜑,𝑚   𝑛,𝐹,𝑥   𝑚,𝑋,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑛)   𝐸(𝑥,𝑚,𝑛)   𝐹(𝑚)   𝑁(𝑚,𝑛)

Proof of Theorem cygznlem3

Dummy variables 𝑎 𝑏 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2727	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		cygzn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2727	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
4		eqid 2727	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
5		cygzn.n	. . . . . 6 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6		hashcl 14339	. . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8		0nn0 12509	. . . . . . . 8 ⊢ 0 ∈ ℕ0
9	8	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
10	7, 9	ifclda 4559	. . . . . 6 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
11	5, 10	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
12		cygzn.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
13	12	zncrng 21465	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
14		crngring 20176	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15		ringgrp 20169	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
16	11, 13, 14, 15	4syl 19	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Grp)
17		cygzn.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
18		cyggrp 19836	. . . . 5 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
19	17, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
20		cygzn.m	. . . . 5 ⊢ · = (.g‘𝐺)
21		cygzn.l	. . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑌)
22		cygzn.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
23		cygzn.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
24		cygzn.f	. . . . 5 ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)
25	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2a 21488	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)
26	12, 1, 21	znzrhfo 21468	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌))
27	11, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
28		foelrn 7111	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
29	27, 28	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
30		foelrn 7111	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
31	27, 30	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
32	29, 31	anim12dan 618	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
33		reeanv 3221	. . . . . . 7 ⊢ (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
34	19	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
37	2, 20, 22	iscyggen 19826	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
38	37	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
39	23, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋 ∈ 𝐵)
41	2, 20, 4	mulgdir 19052	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋)))
42	34, 35, 36, 40, 41	syl13anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋)))
43	11, 13	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ CRing)
44	21	zrhrhm 21424	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45		rhmghm 20412	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌))
46	43, 14, 44, 45	4syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ (ℤring GrpHom 𝑌))
47	46	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌))
48		zringbas 21366	. . . . . . . . . . . . . 14 ⊢ ℤ = (Base‘ℤring)
49		zringplusg 21367	. . . . . . . . . . . . . 14 ⊢ + = (+g‘ℤring)
50	48, 49, 3	ghmlin 19166	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
51	47, 35, 36, 50	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
52	51	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
53		zaddcl 12624	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
54	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
55	53, 54	sylan2 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
56	52, 55	eqtr3d 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝑖 + 𝑗) · 𝑋))
57	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
58	57	adantrr 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
59	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
60	59	adantrl 715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
61	58, 60	oveq12d 7432	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋)))
62	42, 56, 61	3eqtr4d 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))))
63		oveq12 7423	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎(+g‘𝑌)𝑏) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
64	63	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
65		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐿‘𝑖) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑖)))
66		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐿‘𝑗) → (𝐹‘𝑏) = (𝐹‘(𝐿‘𝑗)))
67	65, 66	oveqan12d 7433	. . . . . . . . . 10 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))))
68	64, 67	eqeq12d 2743	. . . . . . . . 9 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)) ↔ (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗)))))
69	62, 68	syl5ibrcom 246	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏))))
70	69	rexlimdvva 3206	. . . . . . 7 ⊢ (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏))))
71	33, 70	biimtrrid 242	. . . . . 6 ⊢ (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏))))
72	71	imp 406	. . . . 5 ⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)))
73	32, 72	syldan 590	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)))
74	1, 2, 3, 4, 16, 19, 25, 73	isghmd 19170	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpHom 𝐺))
75	58, 60	eqeq12d 2743	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
76	2, 5, 12, 20, 21, 22, 17, 23	cygznlem1 21487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐿‘𝑖) = (𝐿‘𝑗) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
77	75, 76	bitr4d 282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) ↔ (𝐿‘𝑖) = (𝐿‘𝑗)))
78	77	biimpd 228	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) → (𝐿‘𝑖) = (𝐿‘𝑗)))
79	65, 66	eqeqan12d 2741	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗))))
80		eqeq12 2744	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎 = 𝑏 ↔ (𝐿‘𝑖) = (𝐿‘𝑗)))
81	79, 80	imbi12d 344	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏) ↔ ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) → (𝐿‘𝑖) = (𝐿‘𝑗))))
82	78, 81	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
83	82	rexlimdvva 3206	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
84	33, 83	biimtrrid 242	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
85	84	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
86	32, 85	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
87	86	ralrimivva 3195	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
88		dff13 7259	. . . . 5 ⊢ (𝐹:(Base‘𝑌)–1-1→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
89	25, 87, 88	sylanbrc 582	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1→𝐵)
90	2, 20, 22	iscyggen2 19827	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
91	19, 90	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
92	23, 91	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)))
93	92	simprd 495	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94		oveq1 7421	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
95	94	eqeq2d 2738	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
96	95	cbvrexvw 3230	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
97	27	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98		fof 6805	. . . . . . . . . . . . 13 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
99	97, 98	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ⟶(Base‘𝑌))
100	99	ffvelcdmda 7088	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿‘𝑗) ∈ (Base‘𝑌))
101	59	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
102	101	eqcomd 2733	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗)))
103		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐿‘𝑗) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑗)))
104	103	rspceeqv 3629	. . . . . . . . . . 11 ⊢ (((𝐿‘𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
105	100, 102, 104	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
106		eqeq1 2731	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹‘𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘𝑎)))
107	106	rexbidv 3173	. . . . . . . . . 10 ⊢ (𝑧 = (𝑗 · 𝑋) → (∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎) ↔ ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎)))
108	105, 107	syl5ibrcom 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
109	108	rexlimdva 3150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
110	96, 109	biimtrid 241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
111	110	ralimdva 3162	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
112	93, 111	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))
113		dffo3 7106	. . . . 5 ⊢ (𝐹:(Base‘𝑌)–onto→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
114	25, 112, 113	sylanbrc 582	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–onto→𝐵)
115		df-f1o 6549	. . . 4 ⊢ (𝐹:(Base‘𝑌)–1-1-onto→𝐵 ↔ (𝐹:(Base‘𝑌)–1-1→𝐵 ∧ 𝐹:(Base‘𝑌)–onto→𝐵))
116	89, 114, 115	sylanbrc 582	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1-onto→𝐵)
117	1, 2	isgim 19207	. . 3 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto→𝐵))
118	74, 116, 117	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpIso 𝐺))
119		brgici 19216	. 2 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌 ≃𝑔 𝐺)
120		gicsym 19220	. 2 ⊢ (𝑌 ≃𝑔 𝐺 → 𝐺 ≃𝑔 𝑌)
121	118, 119, 120	3syl 18	1 ⊢ (𝜑 → 𝐺 ≃𝑔 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065  {crab 3427  ifcif 4524  ⟨cop 4630   class class class wbr 5142   ↦ cmpt 5225  ran crn 5673  ⟶wf 6538  –1-1→wf1 6539  –onto→wfo 6540  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7414  Fincfn 8955  0cc0 11130   + caddc 11133  ℕ₀cn0 12494  ℤcz 12580  ♯chash 14313  Basecbs 17171  +_gcplusg 17224  Grpcgrp 18881  ._gcmg 19014   GrpHom cghm 19158   GrpIso cgim 19202   ≃_𝑔 cgic 19203  CycGrpccyg 19823  Ringcrg 20164  CRingccrg 20165   RingHom crh 20397  ℤ_ringczring 21359  ℤRHomczrh 21412  ℤ/nℤczn 21415

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 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208  ax-addf 11209  ax-mulf 11210

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-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-om 7865  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8718  df-ec 8720  df-qs 8724  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-acn 9957  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-rp 12999  df-fz 13509  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-dvds 16223  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-starv 17239  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ds 17246  df-unif 17247  df-0g 17414  df-imas 17481  df-qus 17482  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-mhm 18731  df-grp 18884  df-minusg 18885  df-sbg 18886  df-mulg 19015  df-subg 19069  df-nsg 19070  df-eqg 19071  df-ghm 19159  df-gim 19204  df-gic 19205  df-od 19474  df-cmn 19728  df-abl 19729  df-cyg 19824  df-mgp 20066  df-rng 20084  df-ur 20113  df-ring 20166  df-cring 20167  df-oppr 20262  df-dvdsr 20285  df-rhm 20400  df-subrng 20472  df-subrg 20497  df-lmod 20734  df-lss 20805  df-lsp 20845  df-sra 21047  df-rgmod 21048  df-lidl 21093  df-rsp 21094  df-2idl 21133  df-cnfld 21267  df-zring 21360  df-zrh 21416  df-zn 21419

This theorem is referenced by:  cygzn  21491