Theorem cygznlem3 21719

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 2763	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		cygzn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2763	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
4		eqid 2763	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
5		cygzn.n	. . . . . 6 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6		hashcl 14388	. . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
7	6	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8		0nn0 12514	. . . . . . . 8 ⊢ 0 ∈ ℕ0
9	8	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
10	7, 9	ifclda 4523	. . . . . 6 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
11	5, 10	eqeltrid 2867	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
12		cygzn.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
13	12	zncrng 21694	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
14		crngring 20322	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15		ringgrp 20315	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
16	11, 13, 14, 15	4syl 20	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Grp)
17		cygzn.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
18		cyggrp 19955	. . . . 5 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
19	17, 18	syl 18	. . . 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 21717	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)
26	12, 1, 21	znzrhfo 21697	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌))
27	11, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
28		foelrn 7102	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
29	27, 28	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
30		foelrn 7102	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
31	27, 30	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
32	29, 31	anim12dan 630	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
33		reeanv 3237	. . . . . . 7 ⊢ (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
34	19	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35		simprl 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
37	2, 20, 22	iscyggen 19945	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
38	37	simplbi 501	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
39	23, 38	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
40	39	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋 ∈ 𝐵)
41	2, 20, 4	mulgdir 19167	. . . . . . . . . . 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 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋)))
43	11, 13	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ CRing)
44	21	zrhrhm 21661	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45		rhmghm 20562	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌))
46	43, 14, 44, 45	4syl 20	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ (ℤring GrpHom 𝑌))
47	46	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌))
48		zringbas 21603	. . . . . . . . . . . . . 14 ⊢ ℤ = (Base‘ℤring)
49		zringplusg 21604	. . . . . . . . . . . . . 14 ⊢ + = (+g‘ℤring)
50	48, 49, 3	ghmlin 19286	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
51	47, 35, 36, 50	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
52	51	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
53		zaddcl 12629	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
54	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
55	53, 54	sylan2 604	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
56	52, 55	eqtr3d 2800	. . . . . . . . . 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 21718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
58	57	adantrr 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
59	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
60	59	adantrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
61	58, 60	oveq12d 7428	. . . . . . . . . 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 2808	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))))
63		oveq12 7419	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎(+g‘𝑌)𝑏) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
64	63	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
65		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐿‘𝑖) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑖)))
66		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐿‘𝑗) → (𝐹‘𝑏) = (𝐹‘(𝐿‘𝑗)))
67	65, 66	oveqan12d 7429	. . . . . . . . . 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 2779	. . . . . . . . 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 250	. . . . . . . 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 3222	. . . . . . 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 246	. . . . . 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 411	. . . . 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 602	. . . 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 19290	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpHom 𝐺))
75	58, 60	eqeq12d 2779	. . . . . . . . . . . . 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 21716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐿‘𝑖) = (𝐿‘𝑗) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
77	75, 76	bitr4d 285	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) ↔ (𝐿‘𝑖) = (𝐿‘𝑗)))
78	77	biimpd 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) → (𝐿‘𝑖) = (𝐿‘𝑗)))
79	65, 66	eqeqan12d 2777	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗))))
80		eqeq12 2780	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎 = 𝑏 ↔ (𝐿‘𝑖) = (𝐿‘𝑗)))
81	79, 80	imbi12d 347	. . . . . . . . . . 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 250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
83	82	rexlimdvva 3222	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
84	33, 83	biimtrrid 246	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
85	84	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
86	32, 85	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
87	86	ralrimivva 3208	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
88		dff13 7252	. . . . 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 594	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1→𝐵)
90	2, 20, 22	iscyggen2 19946	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
91	19, 90	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
92	23, 91	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)))
93	92	simprd 500	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
95	94	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
96	95	cbvrexvw 3244	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
97	27	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98		fof 6792	. . . . . . . . . . . . 13 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
99	97, 98	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ⟶(Base‘𝑌))
100	99	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿‘𝑗) ∈ (Base‘𝑌))
101	59	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
102	101	eqcomd 2769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗)))
103		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐿‘𝑗) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑗)))
104	103	rspceeqv 3604	. . . . . . . . . . 11 ⊢ (((𝐿‘𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
105	100, 102, 104	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
106		eqeq1 2767	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹‘𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘𝑎)))
107	106	rexbidv 3189	. . . . . . . . . 10 ⊢ (𝑧 = (𝑗 · 𝑋) → (∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎) ↔ ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎)))
108	105, 107	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
109	108	rexlimdva 3166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
110	96, 109	biimtrid 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
111	110	ralimdva 3177	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
112	93, 111	mpd 16	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))
113		dffo3 7097	. . . . 5 ⊢ (𝐹:(Base‘𝑌)–onto→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
114	25, 112, 113	sylanbrc 594	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–onto→𝐵)
115		df-f1o 6543	. . . 4 ⊢ (𝐹:(Base‘𝑌)–1-1-onto→𝐵 ↔ (𝐹:(Base‘𝑌)–1-1→𝐵 ∧ 𝐹:(Base‘𝑌)–onto→𝐵))
116	89, 114, 115	sylanbrc 594	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1-onto→𝐵)
117	1, 2	isgim 19327	. . 3 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto→𝐵))
118	74, 116, 117	sylanbrc 594	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpIso 𝐺))
119		brgici 19336	. 2 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌 ≃𝑔 𝐺)
120		gicsym 19340	. 2 ⊢ (𝑌 ≃𝑔 𝐺 → 𝐺 ≃𝑔 𝑌)
121	118, 119, 120	3syl 19	1 ⊢ (𝜑 → 𝐺 ≃𝑔 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416  ifcif 4487  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192  ran crn 5662  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  0cc0 11095   + caddc 11098  ℕ₀cn0 12499  ℤcz 12586  ♯chash 14362  Basecbs 17264  +_gcplusg 17305  Grpcgrp 18995  ._gcmg 19128   GrpHom cghm 19278   GrpIso cgim 19322   ≃_𝑔 cgic 19323  CycGrpccyg 19942  Ringcrg 20310  CRingccrg 20311   RingHom crh 20547  ℤ_ringczring 21596  ℤRHomczrh 21649  ℤ/nℤczn 21652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174  ax-mulf 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-omul 8454  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-acn 9924  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-fz 13531  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-dvds 16306  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-0g 17489  df-imas 17557  df-qus 17558  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-nsg 19185  df-eqg 19186  df-ghm 19279  df-gim 19324  df-gic 19325  df-od 19593  df-cmn 19847  df-abl 19848  df-cyg 19943  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-oppr 20415  df-dvdsr 20435  df-rhm 20550  df-subrng 20645  df-subrg 20669  df-lmod 20983  df-lss 21053  df-lsp 21093  df-sra 21294  df-rgmod 21295  df-lidl 21332  df-rsp 21333  df-2idl 21389  df-cnfld 21523  df-zring 21597  df-zrh 21653  df-zn 21656

This theorem is referenced by:  cygzn  21720