Theorem ghmqusker 19305

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 19304	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
7		ghmgrp1 19236	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
8	2, 7	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ Grp)
9	8	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐺 ∈ Grp)
10	1	ghmker 19260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
11	2, 10	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
12	3, 11	eqeltrid 2845	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
13		nsgsubg 19176	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
14	12, 13	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
15	14	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐾 ∈ (SubGrp‘𝐺))
16		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝐻) = (Base‘𝐻)
18	16, 17	ghmf 19238	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
19	2, 18	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
20	19	ffnd 6737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
21	20	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 Fn (Base‘𝐺))
22	21	adantr 480	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
25		ovexd 7466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
26	23, 24, 25, 8	qusbas 17590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
27		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
28	16, 27	eqger 19196	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
29	12, 13, 28	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
30	29	qsss 8818	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
31	26, 30	eqsstrrd 4019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
32	31	sselda 3983	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
33	32	elpwid 4609	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
34	33	sselda 3983	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) → 𝑥 ∈ (Base‘𝐺))
35	34	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
36	35	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ (Base‘𝐺))
37		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
38	37	eqeq1d 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ((𝐽‘𝑟) = 0 ↔ (𝐹‘𝑥) = 0 ))
39	38	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → (𝐹‘𝑥) = 0 )
40		fniniseg 7080	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝐺) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹‘𝑥) = 0 )))
41	40	biimpar 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝐺) ∧ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (◡𝐹 “ { 0 }))
42	22, 36, 39, 41	syl12anc 837	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ (◡𝐹 “ { 0 }))
43	42, 3	eleqtrrdi 2852	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ 𝐾)
44	27	eqg0el 19201	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ([𝑥](𝐺 ~QG 𝐾) = 𝐾 ↔ 𝑥 ∈ 𝐾))
45	44	biimpar 477	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝐾) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
46	9, 15, 43, 45	syl21anc 838	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
47	29	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
48		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
49	26	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
50	48, 49	eleqtrrd 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
51	50	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
52		simpllr 776	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ 𝑟)
53		qsel 8836	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
55		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐺) = (0g‘𝐺)
56	16, 27, 55	eqgid 19198	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = 𝐾)
57	14, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [(0g‘𝐺)](𝐺 ~QG 𝐾) = 𝐾)
58	57	ad4antr 732	. . . . . . . . . . . . 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 2787	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾))
60	4, 55	qus0 19207	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
61	12, 60	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
62	61	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
63	62	adantr 480	. . . . . . . . . . . 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 2777	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = (0g‘𝑄))
65	62	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾) ↔ 𝑟 = (0g‘𝑄)))
66	65	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → 𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾))
67	66	fveq2d 6910	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
69	68	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
70	16, 55	grpidcl 18983	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
71	8, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ (Base‘𝐺))
72	71	ad4antr 732	. . . . . . . . . . . . 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 19301	. . . . . . . . . . . 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 19240	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g‘𝐺)) = 0 )
75	2, 74	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = 0 )
76	75	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐹‘(0g‘𝐺)) = 0 )
77	67, 73, 76	3eqtrd 2781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐽‘𝑟) = 0 )
78	64, 77	impbida 801	. . . . . . . . . 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 19303	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
80	78, 79	r19.29a 3162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ((𝐽‘𝑟) = 0 ↔ 𝑟 = (0g‘𝑄)))
81	80	pm5.32da 579	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 ) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g‘𝑄))))
82		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → 𝑟 = (0g‘𝑄))
83	4	qusgrp 19204	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
84	12, 83	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ Grp)
85		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
86		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑄) = (0g‘𝑄)
87	85, 86	grpidcl 18983	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ Grp → (0g‘𝑄) ∈ (Base‘𝑄))
88	84, 87	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝑄) ∈ (Base‘𝑄))
89	88	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → (0g‘𝑄) ∈ (Base‘𝑄))
90	82, 89	eqeltrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
91	90	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 = (0g‘𝑄) → 𝑟 ∈ (Base‘𝑄)))
92	91	pm4.71rd 562	. . . . . . . 8 ⊢ (𝜑 → (𝑟 = (0g‘𝑄) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g‘𝑄))))
93	81, 92	bitr4d 282	. . . . . . 7 ⊢ (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 ) ↔ 𝑟 = (0g‘𝑄)))
94	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
95	94	imaexd 7938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
96	95	uniexd 7762	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → ∪ (𝐹 “ 𝑞) ∈ V)
97	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)))
98	21, 35	fnfvelrnd 7102	. . . . . . . . . . . . 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 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ran 𝐹 = (Base‘𝐻))
101	98, 100	eleqtrd 2843	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
102	37, 101	eqeltrd 2841	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
103	102, 79	r19.29a 3162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
104	96, 97, 103	fmpt2d 7144	. . . . . . . . 9 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶(Base‘𝐻))
105	104	ffnd 6737	. . . . . . . 8 ⊢ (𝜑 → 𝐽 Fn (Base‘𝑄))
106		fniniseg 7080	. . . . . . . 8 ⊢ (𝐽 Fn (Base‘𝑄) → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ (𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 )))
107	105, 106	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ (𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 )))
108		velsn 4642	. . . . . . . 8 ⊢ (𝑟 ∈ {(0g‘𝑄)} ↔ 𝑟 = (0g‘𝑄))
109	108	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ {(0g‘𝑄)} ↔ 𝑟 = (0g‘𝑄)))
110	93, 107, 109	3bitr4d 311	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ 𝑟 ∈ {(0g‘𝑄)}))
111	110	eqrdv 2735	. . . . 5 ⊢ (𝜑 → (◡𝐽 “ { 0 }) = {(0g‘𝑄)})
112	85, 17, 86, 1	kerf1ghm 19265	. . . . . 6 ⊢ (𝐽 ∈ (𝑄 GrpHom 𝐻) → (𝐽:(Base‘𝑄)–1-1→(Base‘𝐻) ↔ (◡𝐽 “ { 0 }) = {(0g‘𝑄)}))
113	112	biimpar 477	. . . . 5 ⊢ ((𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ (◡𝐽 “ { 0 }) = {(0g‘𝑄)}) → 𝐽:(Base‘𝑄)–1-1→(Base‘𝐻))
114	6, 111, 113	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1→(Base‘𝐻))
115		f1f1orn 6859	. . . 4 ⊢ (𝐽:(Base‘𝑄)–1-1→(Base‘𝐻) → 𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽)
116	114, 115	syl 17	. . 3 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽)
117		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
118		ovex 7464	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝐾) ∈ V
119	118	ecelqsi 8813	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Base‘𝐺) → [𝑥](𝐺 ~QG 𝐾) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
120	117, 119	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
121	26	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
122	120, 121	eleqtrd 2843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ (Base‘𝑄))
123		elqsi 8810	. . . . . . . . 9 ⊢ (𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) → ∃𝑥 ∈ (Base‘𝐺)𝑟 = [𝑥](𝐺 ~QG 𝐾))
124	50, 123	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ (Base‘𝐺)𝑟 = [𝑥](𝐺 ~QG 𝐾))
125		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
126	125	fveq2d 6910	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐽‘[𝑥](𝐺 ~QG 𝐾)))
127	2	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
128	1, 127, 3, 4, 5, 117	ghmquskerlem1 19301	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥))
129	128	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥))
130	126, 129	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐹‘𝑥))
131	130	3impa 1110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐹‘𝑥))
132	131	eqeq1d 2739	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → ((𝐽‘𝑟) = 𝑦 ↔ (𝐹‘𝑥) = 𝑦))
133	122, 124, 132	rexxfrd2 5413	. . . . . . 7 ⊢ (𝜑 → (∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
134		fvelrnb 6969	. . . . . . . 8 ⊢ (𝐽 Fn (Base‘𝑄) → (𝑦 ∈ ran 𝐽 ↔ ∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦))
135	105, 134	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran 𝐽 ↔ ∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦))
136		fvelrnb 6969	. . . . . . . 8 ⊢ (𝐹 Fn (Base‘𝐺) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
137	20, 136	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
138	133, 135, 137	3bitr4rd 312	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐽))
139	138	eqrdv 2735	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐽)
140	139, 99	eqtr3d 2779	. . . 4 ⊢ (𝜑 → ran 𝐽 = (Base‘𝐻))
141	140	f1oeq3d 6845	. . 3 ⊢ (𝜑 → (𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽 ↔ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
142	116, 141	mpbid 232	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
143	85, 17	isgim 19280	. 2 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) ↔ (𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
144	6, 142, 143	sylanbrc 583	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ^◡ccnv 5684  ran crn 5686   “ cima 5688   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743   / cqs 8744  Basecbs 17247  0_gc0g 17484   /_s cqus 17550  Grpcgrp 18951  SubGrpcsubg 19138  NrmSGrpcnsg 19139   ~_QG cqg 19140   GrpHom cghm 19230   GrpIso cgim 19275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17486  df-imas 17553  df-qus 17554  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-gim 19277

This theorem is referenced by:  gicqusker  19306  lmhmqusker  33445  rhmqusker  33454  aks6d1c6lem5  42178