Theorem cygznlem3 21506

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 2731	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		cygzn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2731	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
4		eqid 2731	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
5		cygzn.n	. . . . . 6 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6		hashcl 14263	. . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8		0nn0 12396	. . . . . . . 8 ⊢ 0 ∈ ℕ0
9	8	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
10	7, 9	ifclda 4508	. . . . . 6 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
11	5, 10	eqeltrid 2835	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
12		cygzn.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
13	12	zncrng 21481	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
14		crngring 20163	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15		ringgrp 20156	. . . . 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 19802	. . . . 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 21504	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)
26	12, 1, 21	znzrhfo 21484	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌))
27	11, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
28		foelrn 7040	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
29	27, 28	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
30		foelrn 7040	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
31	27, 30	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
32	29, 31	anim12dan 619	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
33		reeanv 3204	. . . . . . 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 19792	. . . . . . . . . . . . . 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 19019	. . . . . . . . . . 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 1374	. . . . . . . . . 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 21448	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45		rhmghm 20401	. . . . . . . . . . . . . . 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 21390	. . . . . . . . . . . . . 14 ⊢ ℤ = (Base‘ℤring)
49		zringplusg 21391	. . . . . . . . . . . . . 14 ⊢ + = (+g‘ℤring)
50	48, 49, 3	ghmlin 19133	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
51	47, 35, 36, 50	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
52	51	fveq2d 6826	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
53		zaddcl 12512	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
54	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21505	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
55	53, 54	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
56	52, 55	eqtr3d 2768	. . . . . . . . . 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 21505	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
58	57	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
59	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21505	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
60	59	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
61	58, 60	oveq12d 7364	. . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))))
63		oveq12 7355	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎(+g‘𝑌)𝑏) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))
64	63	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
65		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐿‘𝑖) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑖)))
66		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐿‘𝑗) → (𝐹‘𝑏) = (𝐹‘(𝐿‘𝑗)))
67	65, 66	oveqan12d 7365	. . . . . . . . . 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 2747	. . . . . . . . 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 247	. . . . . . . 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 3189	. . . . . . 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 243	. . . . . 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 591	. . . 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 19137	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpHom 𝐺))
75	58, 60	eqeq12d 2747	. . . . . . . . . . . . 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 21503	. . . . . . . . . . . . 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 229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) → (𝐿‘𝑖) = (𝐿‘𝑗)))
79	65, 66	eqeqan12d 2745	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗))))
80		eqeq12 2748	. . . . . . . . . . . 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 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
83	82	rexlimdvva 3189	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
84	33, 83	biimtrrid 243	. . . . . . . 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 591	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
87	86	ralrimivva 3175	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
88		dff13 7188	. . . . 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 583	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1→𝐵)
90	2, 20, 22	iscyggen2 19793	. . . . . . . . 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 232	. . . . . . 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 7353	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
95	94	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
96	95	cbvrexvw 3211	. . . . . . . 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 6735	. . . . . . . . . . . . 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 7017	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿‘𝑗) ∈ (Base‘𝑌))
101	59	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
102	101	eqcomd 2737	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗)))
103		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐿‘𝑗) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑗)))
104	103	rspceeqv 3595	. . . . . . . . . . 11 ⊢ (((𝐿‘𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
105	100, 102, 104	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
106		eqeq1 2735	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹‘𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘𝑎)))
107	106	rexbidv 3156	. . . . . . . . . 10 ⊢ (𝑧 = (𝑗 · 𝑋) → (∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎) ↔ ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎)))
108	105, 107	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
109	108	rexlimdva 3133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
110	96, 109	biimtrid 242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
111	110	ralimdva 3144	. . . . . 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 7035	. . . . 5 ⊢ (𝐹:(Base‘𝑌)–onto→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
114	25, 112, 113	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–onto→𝐵)
115		df-f1o 6488	. . . 4 ⊢ (𝐹:(Base‘𝑌)–1-1-onto→𝐵 ↔ (𝐹:(Base‘𝑌)–1-1→𝐵 ∧ 𝐹:(Base‘𝑌)–onto→𝐵))
116	89, 114, 115	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1-onto→𝐵)
117	1, 2	isgim 19174	. . 3 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto→𝐵))
118	74, 116, 117	sylanbrc 583	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpIso 𝐺))
119		brgici 19183	. 2 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌 ≃𝑔 𝐺)
120		gicsym 19187	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395  ifcif 4472  ⟨cop 4579   class class class wbr 5089   ↦ cmpt 5170  ran crn 5615  ⟶wf 6477  –1-1→wf1 6478  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  0cc0 11006   + caddc 11009  ℕ₀cn0 12381  ℤcz 12468  ♯chash 14237  Basecbs 17120  +_gcplusg 17161  Grpcgrp 18846  ._gcmg 18980   GrpHom cghm 19124   GrpIso cgim 19169   ≃_𝑔 cgic 19170  CycGrpccyg 19789  Ringcrg 20151  CRingccrg 20152   RingHom crh 20387  ℤ_ringczring 21383  ℤRHomczrh 21436  ℤ/nℤczn 21439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084  ax-addf 11085  ax-mulf 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-ec 8624  df-qs 8628  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-card 9832  df-acn 9835  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-rp 12891  df-fz 13408  df-fl 13696  df-mod 13774  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-dvds 16164  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-0g 17345  df-imas 17412  df-qus 17413  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-grp 18849  df-minusg 18850  df-sbg 18851  df-mulg 18981  df-subg 19036  df-nsg 19037  df-eqg 19038  df-ghm 19125  df-gim 19171  df-gic 19172  df-od 19440  df-cmn 19694  df-abl 19695  df-cyg 19790  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153  df-cring 20154  df-oppr 20255  df-dvdsr 20275  df-rhm 20390  df-subrng 20461  df-subrg 20485  df-lmod 20795  df-lss 20865  df-lsp 20905  df-sra 21107  df-rgmod 21108  df-lidl 21145  df-rsp 21146  df-2idl 21187  df-cnfld 21292  df-zring 21384  df-zrh 21440  df-zn 21443

This theorem is referenced by:  cygzn  21507