Theorem cygznlem3 21785

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 2760	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		cygzn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2760	. . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌)
4		eqid 2760	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
5		cygzn.n	. . . . . 6 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6		hashcl 14423	. . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
7	6	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8		0nn0 12546	. . . . . . . 8 ⊢ 0 ∈ ℕ0
9	8	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
10	7, 9	ifclda 4518	. . . . . 6 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
11	5, 10	eqeltrid 2864	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
12		cygzn.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
13	12	zncrng 21760	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
14		crngring 20387	. . . . 5 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15		ringgrp 20380	. . . . 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 20020	. . . . 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 21783	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)
26	12, 1, 21	znzrhfo 21763	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→(Base‘𝑌))
27	11, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
28		foelrn 7101	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
29	27, 28	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖))
30		foelrn 7101	. . . . . . 7 ⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
31	27, 30	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))
32	29, 31	anim12dan 631	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
33		reeanv 3234	. . . . . . 7 ⊢ (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)))
34	19	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
37	2, 20, 22	iscyggen 20010	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
38	37	simplbi 502	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
39	23, 38	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
40	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋 ∈ 𝐵)
41	2, 20, 4	mulgdir 19232	. . . . . . . . . . 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 21727	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45		rhmghm 20628	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌))
48		zringbas 21669	. . . . . . . . . . . . . 14 ⊢ ℤ = (Base‘ℤring)
49		zringplusg 21670	. . . . . . . . . . . . . 14 ⊢ + = (+g‘ℤring)
50	48, 49, 3	ghmlin 19351	. . . . . . . . . . . . 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 6883	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
53		zaddcl 12661	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
54	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
55	53, 54	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
56	52, 55	eqtr3d 2797	. . . . . . . . . 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 21784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
58	57	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋))
59	2, 5, 12, 20, 21, 22, 17, 23, 24	cygznlem2 21784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
60	59	adantrl 729	. . . . . . . . . . 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 2805	. . . . . . . . 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 6883	. . . . . . . . . 10 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))))
65		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐿‘𝑖) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑖)))
66		fveq2 6879	. . . . . . . . . . 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 2776	. . . . . . . . 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 3219	. . . . . . 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 412	. . . . 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 603	. . . 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 19355	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpHom 𝐺))
75	58, 60	eqeq12d 2776	. . . . . . . . . . . . 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 21782	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗))))
80		eqeq12 2777	. . . . . . . . . . . 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 3219	. . . . . . . . 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 412	. . . . . . 7 ⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
86	32, 85	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
87	86	ralrimivva 3205	. . . . 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 595	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1→𝐵)
90	2, 20, 22	iscyggen2 20011	. . . . . . . . 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 501	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94		oveq1 7421	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
95	94	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
96	95	cbvrexvw 3241	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
97	27	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98		fof 6790	. . . . . . . . . . . . 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 7078	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿‘𝑗) ∈ (Base‘𝑌))
101	59	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋))
102	101	eqcomd 2766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗)))
103		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐿‘𝑗) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑗)))
104	103	rspceeqv 3599	. . . . . . . . . . 11 ⊢ (((𝐿‘𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
105	100, 102, 104	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))
106		eqeq1 2764	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹‘𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘𝑎)))
107	106	rexbidv 3186	. . . . . . . . . 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 3163	. . . . . . . 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 3174	. . . . . 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 7096	. . . . 5 ⊢ (𝐹:(Base‘𝑌)–onto→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)))
114	25, 112, 113	sylanbrc 595	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–onto→𝐵)
115		df-f1o 6540	. . . 4 ⊢ (𝐹:(Base‘𝑌)–1-1-onto→𝐵 ↔ (𝐹:(Base‘𝑌)–1-1→𝐵 ∧ 𝐹:(Base‘𝑌)–onto→𝐵))
116	89, 114, 115	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1-onto→𝐵)
117	1, 2	isgim 19392	. . 3 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto→𝐵))
118	74, 116, 117	sylanbrc 595	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpIso 𝐺))
119		brgici 19401	. 2 ⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌 ≃𝑔 𝐺)
120		gicsym 19405	. 2 ⊢ (𝑌 ≃𝑔 𝐺 → 𝐺 ≃𝑔 𝑌)
121	118, 119, 120	3syl 19	1 ⊢ (𝜑 → 𝐺 ≃𝑔 𝑌)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  ifcif 4482  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186  ran crn 5656  ⟶wf 6529  –1-1→wf1 6530  –onto→wfo 6531  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7414  Fincfn 8955  0cc0 11127   + caddc 11130  ℕ₀cn0 12531  ℤcz 12618  ♯chash 14397  Basecbs 17304  +_gcplusg 17345  Grpcgrp 19060  ._gcmg 19193   GrpHom cghm 19343   GrpIso cgim 19387   ≃_𝑔 cgic 19388  CycGrpccyg 20007  Ringcrg 20375  CRingccrg 20376   RingHom crh 20613  ℤ_ringczring 21662  ℤRHomczrh 21715  ℤ/nℤczn 21718

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205  ax-addf 11206  ax-mulf 11207

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-oadd 8462  df-omul 8463  df-er 8699  df-ec 8701  df-qs 8705  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-oi 9485  df-card 9947  df-acn 9950  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-rp 13046  df-fz 13565  df-fl 13856  df-mod 13934  df-seq 14069  df-exp 14129  df-hash 14398  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-dvds 16346  df-struct 17242  df-sets 17259  df-slot 17277  df-ndx 17289  df-base 17305  df-ress 17326  df-plusg 17358  df-mulr 17359  df-starv 17360  df-sca 17361  df-vsca 17362  df-ip 17363  df-tset 17364  df-ple 17365  df-ds 17367  df-unif 17368  df-0g 17529  df-imas 17597  df-qus 17598  df-mgm 18733  df-sgrp 18824  df-mnd 18840  df-mhm 18894  df-grp 19063  df-minusg 19064  df-sbg 19065  df-mulg 19194  df-subg 19249  df-nsg 19250  df-eqg 19251  df-ghm 19344  df-gim 19389  df-gic 19390  df-od 19658  df-cmn 19912  df-abl 19913  df-cyg 20008  df-mgp 20277  df-rng 20291  df-ur 20324  df-ring 20377  df-cring 20378  df-oppr 20481  df-dvdsr 20501  df-rhm 20616  df-subrng 20711  df-subrg 20735  df-lmod 21049  df-lss 21119  df-lsp 21159  df-sra 21360  df-rgmod 21361  df-lidl 21398  df-rsp 21399  df-2idl 21455  df-cnfld 21589  df-zring 21663  df-zrh 21719  df-zn 21722

This theorem is used by:  cygzn  21786