Theorem ghmqusker 19552

Description: A surjective group homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 15-Feb-2025.)

Hypotheses

Ref	Expression
ghmqusker.1	⊢ 0 = (0g‘𝐻)
ghmqusker.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusker.k	⊢ 𝐾 = (◡𝐹 “ { 0 })
ghmqusker.q	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
ghmqusker.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
ghmqusker.s	⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))

Assertion

Ref	Expression
ghmqusker	⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))

Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑄,𝑞   𝜑,𝑞

Allowed substitution hint:   0 (𝑞)

Proof of Theorem ghmqusker

Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmqusker.1	. . 3 ⊢ 0 = (0g‘𝐻)
2		ghmqusker.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
3		ghmqusker.k	. . 3 ⊢ 𝐾 = (◡𝐹 “ { 0 })
4		ghmqusker.q	. . 3 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
5		ghmqusker.j	. . 3 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
6	1, 2, 3, 4, 5	ghmquskerlem3 19551	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
7		ghmgrp1 19482	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
8	2, 7	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ Grp)
9	8	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐺 ∈ Grp)
10	1	ghmker 19506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
11	2, 10	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
12	3, 11	eqeltrid 2868	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
13		nsgsubg 19418	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
14	12, 13	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
15	14	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐾 ∈ (SubGrp‘𝐺))
16		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝐻) = (Base‘𝐻)
18	16, 17	ghmf 19484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
19	2, 18	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
20	19	ffnd 6740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
21	20	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 Fn (Base‘𝐺))
22	21	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐹 Fn (Base‘𝐺))
23	4	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
24		eqidd 2764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
25		ovexd 7486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
26	23, 24, 25, 8	qusbas 17767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
27		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
28	16, 27	eqger 19440	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
29	12, 13, 28	3syl 19	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
30	29	qsss 8834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
31	26, 30	eqsstrrd 3948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
32	31	sselda 3913	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
33	32	elpwid 4569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
34	33	sselda 3913	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) → 𝑥 ∈ (Base‘𝐺))
35	34	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
36	35	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ (Base‘𝐺))
37		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
38	37	eqeq1d 2765	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ((𝐽‘𝑟) = 0 ↔ (𝐹‘𝑥) = 0 ))
39	38	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → (𝐹‘𝑥) = 0 )
40		fniniseg 7091	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝐺) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹‘𝑥) = 0 )))
41	40	biimpar 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝐺) ∧ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (◡𝐹 “ { 0 }))
42	22, 36, 39, 41	syl12anc 850	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ (◡𝐹 “ { 0 }))
43	42, 3	eleqtrrdi 2875	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ 𝐾)
44	27	eqg0el 19448	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ([𝑥](𝐺 ~QG 𝐾) = 𝐾 ↔ 𝑥 ∈ 𝐾))
45	44	biimpar 483	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝐾) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
46	9, 15, 43, 45	syl21anc 851	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
47	29	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
48		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
49	26	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
50	48, 49	eleqtrrd 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
51	50	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
52		simpllr 788	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ 𝑟)
53		qsel 8855	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥 ∈ 𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
54	47, 51, 52, 53	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
55		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐺) = (0g‘𝐺)
56	16, 27, 55	eqgid 19442	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = 𝐾)
57	14, 56	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [(0g‘𝐺)](𝐺 ~QG 𝐾) = 𝐾)
58	57	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = 𝐾)
59	46, 54, 58	3eqtr4d 2808	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾))
60	4, 55	qus0 19454	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
61	12, 60	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
62	61	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
63	62	adantr 486	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
64	59, 63	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = (0g‘𝑄))
65	62	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾) ↔ 𝑟 = (0g‘𝑄)))
66	65	biimpar 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → 𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾))
67	66	fveq2d 6920	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐽‘𝑟) = (𝐽‘[(0g‘𝐺)](𝐺 ~QG 𝐾)))
68	2	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
69	68	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
70	16, 55	grpidcl 19226	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
71	8, 70	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ (Base‘𝐺))
72	71	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (0g‘𝐺) ∈ (Base‘𝐺))
73	1, 69, 3, 4, 5, 72	ghmquskerlem1 19548	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐽‘[(0g‘𝐺)](𝐺 ~QG 𝐾)) = (𝐹‘(0g‘𝐺)))
74	55, 1	ghmid 19486	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g‘𝐺)) = 0 )
75	2, 74	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = 0 )
76	75	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐹‘(0g‘𝐺)) = 0 )
77	67, 73, 76	3eqtrd 2802	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐽‘𝑟) = 0 )
78	64, 77	impbida 813	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ((𝐽‘𝑟) = 0 ↔ 𝑟 = (0g‘𝑄)))
79	1, 68, 3, 4, 5, 48	ghmquskerlem2 19550	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
80	78, 79	r19.29a 3174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ((𝐽‘𝑟) = 0 ↔ 𝑟 = (0g‘𝑄)))
81	80	pm5.32da 590	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 ) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g‘𝑄))))
82		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → 𝑟 = (0g‘𝑄))
83	4	qusgrp 19451	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
84	12, 83	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ Grp)
85		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
86		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑄) = (0g‘𝑄)
87	85, 86	grpidcl 19226	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ Grp → (0g‘𝑄) ∈ (Base‘𝑄))
88	84, 87	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝑄) ∈ (Base‘𝑄))
89	88	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → (0g‘𝑄) ∈ (Base‘𝑄))
90	82, 89	eqeltrd 2864	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
91	90	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 = (0g‘𝑄) → 𝑟 ∈ (Base‘𝑄)))
92	91	pm4.71rd 572	. . . . . . . 8 ⊢ (𝜑 → (𝑟 = (0g‘𝑄) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g‘𝑄))))
93	81, 92	bitr4d 285	. . . . . . 7 ⊢ (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 ) ↔ 𝑟 = (0g‘𝑄)))
94	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
95	94	imaexd 7962	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
96	95	uniexd 7791	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → ∪ (𝐹 “ 𝑞) ∈ V)
97	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)))
98	21, 35	fnfvelrnd 7114	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ ran 𝐹)
99		ghmqusker.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))
100	99	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ran 𝐹 = (Base‘𝐻))
101	98, 100	eleqtrd 2866	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
102	37, 101	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
103	102, 79	r19.29a 3174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
104	96, 97, 103	fmpt2d 7157	. . . . . . . . 9 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶(Base‘𝐻))
105	104	ffnd 6740	. . . . . . . 8 ⊢ (𝜑 → 𝐽 Fn (Base‘𝑄))
106		fniniseg 7091	. . . . . . . 8 ⊢ (𝐽 Fn (Base‘𝑄) → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ (𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 )))
107	105, 106	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ (𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 )))
108		velsn 4600	. . . . . . . 8 ⊢ (𝑟 ∈ {(0g‘𝑄)} ↔ 𝑟 = (0g‘𝑄))
109	108	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ {(0g‘𝑄)} ↔ 𝑟 = (0g‘𝑄)))
110	93, 107, 109	3bitr4d 314	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ 𝑟 ∈ {(0g‘𝑄)}))
111	110	eqrdv 2761	. . . . 5 ⊢ (𝜑 → (◡𝐽 “ { 0 }) = {(0g‘𝑄)})
112	85, 17, 86, 1	kerf1ghm 19511	. . . . . 6 ⊢ (𝐽 ∈ (𝑄 GrpHom 𝐻) → (𝐽:(Base‘𝑄)⟶1-1(Base‘𝐻) ↔ (◡𝐽 “ { 0 }) = {(0g‘𝑄)}))
113	112	biimpar 483	. . . . 5 ⊢ ((𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ (◡𝐽 “ { 0 }) = {(0g‘𝑄)}) → 𝐽:(Base‘𝑄)⟶1-1(Base‘𝐻))
114	6, 111, 113	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶1-1(Base‘𝐻))
115		f1f1orn 6866	. . . 4 ⊢ (𝐽:(Base‘𝑄)⟶1-1(Base‘𝐻) → 𝐽:(Base‘𝑄)⟶1-1ontoran 𝐽)
116	114, 115	syl 18	. . 3 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶1-1ontoran 𝐽)
117		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
118		ovex 7484	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝐾) ∈ V
119	118	ecelqsi 8828	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Base‘𝐺) → [𝑥](𝐺 ~QG 𝐾) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
120	117, 119	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
121	26	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
122	120, 121	eleqtrd 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ (Base‘𝑄))
123		elqsi 8824	. . . . . . . . 9 ⊢ (𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) → ∃𝑥 ∈ (Base‘𝐺)𝑟 = [𝑥](𝐺 ~QG 𝐾))
124	50, 123	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ (Base‘𝐺)𝑟 = [𝑥](𝐺 ~QG 𝐾))
125		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
126	125	fveq2d 6920	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐽‘[𝑥](𝐺 ~QG 𝐾)))
127	2	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
128	1, 127, 3, 4, 5, 117	ghmquskerlem1 19548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥))
129	128	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥))
130	126, 129	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐹‘𝑥))
131	130	3impa 1127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐹‘𝑥))
132	131	eqeq1d 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → ((𝐽‘𝑟) = 𝑦 ↔ (𝐹‘𝑥) = 𝑦))
133	122, 124, 132	rexxfrd2 5382	. . . . . . 7 ⊢ (𝜑 → (∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
134		fvelrnb 6976	. . . . . . . 8 ⊢ (𝐽 Fn (Base‘𝑄) → (𝑦 ∈ ran 𝐽 ↔ ∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦))
135	105, 134	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran 𝐽 ↔ ∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦))
136		fvelrnb 6976	. . . . . . . 8 ⊢ (𝐹 Fn (Base‘𝐺) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
137	20, 136	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
138	133, 135, 137	3bitr4rd 315	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐽))
139	138	eqrdv 2761	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐽)
140	139, 99	eqtr3d 2800	. . . 4 ⊢ (𝜑 → ran 𝐽 = (Base‘𝐻))
141	140	f1oeq3d 6851	. . 3 ⊢ (𝜑 → (𝐽:(Base‘𝑄)⟶1-1ontoran 𝐽 ↔ 𝐽:(Base‘𝑄)⟶1-1onto(Base‘𝐻)))
142	116, 141	mpbid 235	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶1-1onto(Base‘𝐻))
143	85, 17	isgim 19526	. 2 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) ↔ (𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ 𝐽:(Base‘𝑄)⟶1-1onto(Base‘𝐻)))
144	6, 142, 143	sylanbrc 595	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454  𝒫 cpw 4560  {csn 4588  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5657  ran crn 5659   “ cima 5661   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   Er wer 8752  [cec 8753   / cqs 8754  Basecbs 17431  0_gc0g 17660   /_s cqus 17727  Grpcgrp 19194  SubGrpcsubg 19380  NrmSGrpcnsg 19381   ~_QG cqg 19382   GrpHom cghm 19477   GrpIso cgim 19521

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-ec 8757  df-qs 8761  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-0g 17662  df-imas 17730  df-qus 17731  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-subg 19383  df-nsg 19384  df-eqg 19385  df-ghm 19478  df-gim 19523

This theorem is used by:  gicqusker  19553  lmhmqusker  34072  rhmqusker  34080  aks6d1c6lem5  43493