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Theorem cygznlem3 21507

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 2725	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		cygzn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2725	. . . 4 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
4		eqid 2725	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
5		cygzn.n	. . . . . 6 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6		hashcl 14347	. . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
7	6	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
8		0nn0 12517	. . . . . . . 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 2829	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
12		cygzn.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
13	12	zncrng 21482	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 ∈ CRing)
14		crngring 20189	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15		ringgrp 20182	. . . . 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 19849	. . . . 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 21505	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)
26	12, 1, 21	znzrhfo 21485	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑌))
27	11, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
28		foelrn 7112	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
29	27, 28	sylan 578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
30		foelrn 7112	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
31	27, 30	sylan 578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
32	29, 31	anim12dan 617	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
33		reeanv 3217	. . . . . . 7 ⊢ (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
34	19	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35		simprl 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36		simprr 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
37	2, 20, 22	iscyggen 19839	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
38	37	simplbi 496	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
39	23, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
40	39	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋 ∈ 𝐵)
41	2, 20, 4	mulgdir 19065	. . . . . . . . . . 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 1369	. . . . . . . . . 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 21441	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑌))
45		rhmghm 20427	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑌) → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
46	43, 14, 44, 45	4syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
47	46	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤ_ring GrpHom 𝑌))
48		zringbas 21383	. . . . . . . . . . . . . 14 ⊢ ℤ = (Base‘ℤ_ring)
49		zringplusg 21384	. . . . . . . . . . . . . 14 ⊢ + = (+_g‘ℤ_ring)
50	48, 49, 3	ghmlin 19179	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (ℤ_ring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+_g‘𝑌)(𝐿‘𝑗)))
51	47, 35, 36, 50	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+_g‘𝑌)(𝐿‘𝑗)))
52	51	fveq2d 6896	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿‘𝑖)(+_g‘𝑌)(𝐿‘𝑗))))
53		zaddcl 12632	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
54	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
55	53, 54	sylan2 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
56	52, 55	eqtr3d 2767	. . . . . . . . . 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 21506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
58	57	adantrr 715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
59	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
60	59	adantrl 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
61	58, 60	oveq12d 7434	. . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+_g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+_g‘𝐺)(𝐹‘(𝐿‘𝑗))))
63		oveq12 7425	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎(+_g‘𝑌)𝑏) = ((𝐿‘𝑖)(+_g‘𝑌)(𝐿‘𝑗)))
64	63	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+_g‘𝑌)𝑏)) = (𝐹‘((𝐿‘𝑖)(+_g‘𝑌)(𝐿‘𝑗))))
65		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐿‘𝑖) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑖)))
66		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐿‘𝑗) → (𝐹‘𝑏) = (𝐹‘(𝐿‘𝑗)))
67	65, 66	oveqan12d 7435	. . . . . . . . . 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 2741	. . . . . . . . 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 3202	. . . . . . 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 405	. . . . 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 589	. . . 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 19183	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpHom 𝐺))
75	58, 60	eqeq12d 2741	. . . . . . . . . . . . 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 21504	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐿‘𝑖) = (𝐿‘𝑗) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
77	75, 76	bitr4d 281	. . . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗))))
80		eqeq12 2742	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎 = 𝑏 ↔ (𝐿‘𝑖) = (𝐿‘𝑗)))
81	79, 80	imbi12d 343	. . . . . . . . . . 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 3202	. . . . . . . . 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 405	. . . . . . 7 ⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
86	32, 85	syldan 589	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
87	86	ralrimivva 3191	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
88		dff13 7261	. . . . 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 581	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1→𝐵)
90	2, 20, 22	iscyggen2 19840	. . . . . . . . 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 494	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
95	94	eqeq2d 2736	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
96	95	cbvrexvw 3226	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
97	27	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98		fof 6806	. . . . . . . . . . . . 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 7089	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿‘𝑗) ∈ (Base‘𝑌))
101	59	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
102	101	eqcomd 2731	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗)))
103		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐿‘𝑗) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑗)))
104	103	rspceeqv 3623	. . . . . . . . . . 11 ⊢ (((𝐿‘𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
105	100, 102, 104	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
106		eqeq1 2729	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹‘𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘𝑎)))
107	106	rexbidv 3169	. . . . . . . . . 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 3145	. . . . . . . 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 3157	. . . . . 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 7107	. . . . 5 ⊢ (𝐹:(Base‘𝑌)–onto→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
114	25, 112, 113	sylanbrc 581	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–onto→𝐵)
115		df-f1o 6550	. . . 4 ⊢ (𝐹:(Base‘𝑌)–1-1-onto→𝐵 ↔ (𝐹:(Base‘𝑌)–1-1→𝐵 ∧ 𝐹:(Base‘𝑌)–onto→𝐵))
116	89, 114, 115	sylanbrc 581	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1-onto→𝐵)
117	1, 2	isgim 19220	. . 3 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto→𝐵))
118	74, 116, 117	sylanbrc 581	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpIso 𝐺))
119		brgici 19229	. 2 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌 ≃_𝑔 𝐺)
120		gicsym 19233	. 2 ⊢ (𝑌 ≃_𝑔 𝐺 → 𝐺 ≃_𝑔 𝑌)
121	118, 119, 120	3syl 18	1 ⊢ (𝜑 → 𝐺 ≃_𝑔 𝑌)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 {crab 3419 ifcif 4524 ⟨cop 4630 class class class wbr 5143 ↦ cmpt 5226 ran crn 5673 ⟶wf 6539 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 0cc0 11138 + caddc 11141 ℕ₀cn0 12502 ℤcz 12588 ♯chash 14321 Basecbs 17179 +_gcplusg 17232 Grpcgrp 18894 ._gcmg 19027 GrpHom cghm 19171 GrpIso cgim 19215 ≃_𝑔 cgic 19216 CycGrpccyg 19836 Ringcrg 20177 CRingccrg 20178 RingHom crh 20412 ℤ_ringczring 21376 ℤRHomczrh 21429 ℤ/nℤczn 21432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-omul 8490 df-er 8723 df-ec 8725 df-qs 8729 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-imas 17489 df-qus 17490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-nsg 19083 df-eqg 19084 df-ghm 19172 df-gim 19217 df-gic 19218 df-od 19487 df-cmn 19741 df-abl 19742 df-cyg 19837 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-lsp 20860 df-sra 21062 df-rgmod 21063 df-lidl 21108 df-rsp 21109 df-2idl 21148 df-cnfld 21284 df-zring 21377 df-zrh 21433 df-zn 21436

This theorem is referenced by: cygzn 21508

W3C validator