Theorem ghmqusker 19253

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 19252	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
7		ghmgrp1 19184	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
8	2, 7	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ Grp)
9	8	ad4antr 738	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐺 ∈ Grp)
10	1	ghmker 19208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
11	2, 10	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
12	3, 11	eqeltrid 2843	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
13		nsgsubg 19124	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
14	12, 13	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
15	14	ad4antr 738	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝐾 ∈ (SubGrp‘𝐺))
16		eqid 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝐻) = (Base‘𝐻)
18	16, 17	ghmf 19186	. . . . . . . . . . . . . . . . . . . 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 6656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
21	20	ad3antrrr 736	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 Fn (Base‘𝐺))
22	21	adantr 481	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
25		ovexd 7391	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
26	23, 24, 25, 8	qusbas 17500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
27		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
28	16, 27	eqger 19144	. . . . . . . . . . . . . . . . . . . . . . . 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 8712	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
31	26, 30	eqsstrrd 3950	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
32	31	sselda 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
33	32	elpwid 4538	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
34	33	sselda 3915	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) → 𝑥 ∈ (Base‘𝐺))
35	34	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
36	35	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ (Base‘𝐺))
37		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
38	37	eqeq1d 2741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ((𝐽‘𝑟) = 0 ↔ (𝐹‘𝑥) = 0 ))
39	38	biimpa 477	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → (𝐹‘𝑥) = 0 )
40		fniniseg 7001	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝐺) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹‘𝑥) = 0 )))
41	40	biimpar 478	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝐺) ∧ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹‘𝑥) = 0 )) → 𝑥 ∈ (◡𝐹 “ { 0 }))
42	22, 36, 39, 41	syl12anc 842	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ (◡𝐹 “ { 0 }))
43	42, 3	eleqtrrdi 2850	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ 𝐾)
44	27	eqg0el 19149	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ([𝑥](𝐺 ~QG 𝐾) = 𝐾 ↔ 𝑥 ∈ 𝐾))
45	44	biimpar 478	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝐾) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
46	9, 15, 43, 45	syl21anc 843	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
47	29	ad4antr 738	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
48		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
49	26	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
50	48, 49	eleqtrrd 2842	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
51	50	ad3antrrr 736	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
52		simpllr 781	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑥 ∈ 𝑟)
53		qsel 8733	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
55		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐺) = (0g‘𝐺)
56	16, 27, 55	eqgid 19146	. . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾))
60	4, 55	qus0 19155	. . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → [(0g‘𝐺)](𝐺 ~QG 𝐾) = (0g‘𝑄))
63	62	adantr 481	. . . . . . . . . . . 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 2774	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ (𝐽‘𝑟) = 0 ) → 𝑟 = (0g‘𝑄))
65	62	eqeq2d 2750	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾) ↔ 𝑟 = (0g‘𝑄)))
66	65	biimpar 478	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → 𝑟 = [(0g‘𝐺)](𝐺 ~QG 𝐾))
67	66	fveq2d 6831	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
69	68	ad3antrrr 736	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
70	16, 55	grpidcl 18932	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
71	8, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐺) ∈ (Base‘𝐺))
72	71	ad4antr 738	. . . . . . . . . . . . 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 19249	. . . . . . . . . . . 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 19188	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g‘𝐺)) = 0 )
75	2, 74	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = 0 )
76	75	ad4antr 738	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐹‘(0g‘𝐺)) = 0 )
77	67, 73, 76	3eqtrd 2778	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑟 = (0g‘𝑄)) → (𝐽‘𝑟) = 0 )
78	64, 77	impbida 806	. . . . . . . . . 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 19251	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
80	78, 79	r19.29a 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ((𝐽‘𝑟) = 0 ↔ 𝑟 = (0g‘𝑄)))
81	80	pm5.32da 584	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 ) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g‘𝑄))))
82		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → 𝑟 = (0g‘𝑄))
83	4	qusgrp 19152	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
84	12, 83	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ Grp)
85		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
86		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑄) = (0g‘𝑄)
87	85, 86	grpidcl 18932	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ Grp → (0g‘𝑄) ∈ (Base‘𝑄))
88	84, 87	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝑄) ∈ (Base‘𝑄))
89	88	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → (0g‘𝑄) ∈ (Base‘𝑄))
90	82, 89	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 = (0g‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
91	90	ex 413	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 = (0g‘𝑄) → 𝑟 ∈ (Base‘𝑄)))
92	91	pm4.71rd 567	. . . . . . . 8 ⊢ (𝜑 → (𝑟 = (0g‘𝑄) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g‘𝑄))))
93	81, 92	bitr4d 283	. . . . . . 7 ⊢ (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽‘𝑟) = 0 ) ↔ 𝑟 = (0g‘𝑄)))
94	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
95	94	imaexd 7856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
96	95	uniexd 7685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → ∪ (𝐹 “ 𝑞) ∈ V)
97	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)))
98	21, 35	fnfvelrnd 7023	. . . . . . . . . . . . 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 736	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ran 𝐹 = (Base‘𝐻))
101	98, 100	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
102	37, 101	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
103	102, 79	r19.29a 3147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
104	96, 97, 103	fmpt2d 7066	. . . . . . . . 9 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶(Base‘𝐻))
105	104	ffnd 6656	. . . . . . . 8 ⊢ (𝜑 → 𝐽 Fn (Base‘𝑄))
106		fniniseg 7001	. . . . . . . 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 4571	. . . . . . . 8 ⊢ (𝑟 ∈ {(0g‘𝑄)} ↔ 𝑟 = (0g‘𝑄))
109	108	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑟 ∈ {(0g‘𝑄)} ↔ 𝑟 = (0g‘𝑄)))
110	93, 107, 109	3bitr4d 312	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ (◡𝐽 “ { 0 }) ↔ 𝑟 ∈ {(0g‘𝑄)}))
111	110	eqrdv 2737	. . . . 5 ⊢ (𝜑 → (◡𝐽 “ { 0 }) = {(0g‘𝑄)})
112	85, 17, 86, 1	kerf1ghm 19213	. . . . . 6 ⊢ (𝐽 ∈ (𝑄 GrpHom 𝐻) → (𝐽:(Base‘𝑄)–1-1→(Base‘𝐻) ↔ (◡𝐽 “ { 0 }) = {(0g‘𝑄)}))
113	112	biimpar 478	. . . . 5 ⊢ ((𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ (◡𝐽 “ { 0 }) = {(0g‘𝑄)}) → 𝐽:(Base‘𝑄)–1-1→(Base‘𝐻))
114	6, 111, 113	syl2anc 590	. . . 4 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1→(Base‘𝐻))
115		f1f1orn 6778	. . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
118		ovex 7389	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝐾) ∈ V
119	118	ecelqsi 8706	. . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
122	120, 121	eleqtrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ (Base‘𝑄))
123		elqsi 8702	. . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
126	125	fveq2d 6831	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐽‘[𝑥](𝐺 ~QG 𝐾)))
127	2	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
128	1, 127, 3, 4, 5, 117	ghmquskerlem1 19249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥))
129	128	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥))
130	126, 129	eqtrd 2774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐹‘𝑥))
131	130	3impa 1115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑟) = (𝐹‘𝑥))
132	131	eqeq1d 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → ((𝐽‘𝑟) = 𝑦 ↔ (𝐹‘𝑥) = 𝑦))
133	122, 124, 132	rexxfrd2 5342	. . . . . . 7 ⊢ (𝜑 → (∃𝑟 ∈ (Base‘𝑄)(𝐽‘𝑟) = 𝑦 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹‘𝑥) = 𝑦))
134		fvelrnb 6887	. . . . . . . 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 6887	. . . . . . . 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 313	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐽))
139	138	eqrdv 2737	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐽)
140	139, 99	eqtr3d 2776	. . . 4 ⊢ (𝜑 → ran 𝐽 = (Base‘𝐻))
141	140	f1oeq3d 6764	. . 3 ⊢ (𝜑 → (𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽 ↔ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
142	116, 141	mpbid 233	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
143	85, 17	isgim 19228	. 2 ⊢ (𝐽 ∈ (𝑄 GrpIso 𝐻) ↔ (𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
144	6, 142, 143	sylanbrc 589	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431  𝒫 cpw 4529  {csn 4555  ∪ cuni 4838   ↦ cmpt 5153  ^◡ccnv 5617  ran crn 5619   “ cima 5621   Fn wfn 6480  ⟶wf 6481  –1-1→wf1 6482  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356   Er wer 8630  [cec 8631   / cqs 8632  Basecbs 17170  0_gc0g 17393   /_s cqus 17460  Grpcgrp 18900  SubGrpcsubg 19087  NrmSGrpcnsg 19088   ~_QG cqg 19089   GrpHom cghm 19178   GrpIso cgim 19223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-ec 8635  df-qs 8639  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-0g 17395  df-imas 17463  df-qus 17464  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-nsg 19091  df-eqg 19092  df-ghm 19179  df-gim 19225

This theorem is referenced by:  gicqusker  19254  lmhmqusker  33500  rhmqusker  33509  aks6d1c6lem5  42662