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Theorem ghmquskerlem3 19213

Description: The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 22-Mar-2025.)

**Hypotheses**
Ref	Expression
ghmqusker.1	⊢ 0 = (0_g‘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))

**Assertion**
Ref	Expression
ghmquskerlem3	⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 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 ghmquskerlem3 Dummy variables 𝑟 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2734	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2734	. 2 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
4		eqid 2734	. 2 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
5		ghmqusker.k	. . . 4 ⊢ 𝐾 = (^◡𝐹 “ { 0 })
6		ghmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
7		ghmqusker.1	. . . . . 6 ⊢ 0 = (0_g‘𝐻)
8	7	ghmker 19169	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
10	5, 9	eqeltrid 2838	. . 3 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
11		ghmqusker.q	. . . 4 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾))
12	11	qusgrp 19113	. . 3 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
13	10, 12	syl 17	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
14		ghmrn 19156	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻))
15		subgrcl 19059	. . 3 ⊢ (ran 𝐹 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
16	6, 14, 15	3syl 18	. 2 ⊢ (𝜑 → 𝐻 ∈ Grp)
17	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
18	17	imaexd 7856	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
19	18	uniexd 7685	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → ∪ (𝐹 “ 𝑞) ∈ V)
20		ghmqusker.j	. . . 4 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
21	20	a1i 11	. . 3 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)))
22		simpr 484	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
23		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
24	23, 2	ghmf 19147	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
25	6, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
26	25	frnd 6668	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐻))
27	26	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ran 𝐹 ⊆ (Base‘𝐻))
28	25	ffnd 6661	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
29	28	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 Fn (Base‘𝐺))
30	11	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)))
31		eqidd 2735	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
32		ovexd 7391	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
33		ghmgrp1 19145	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
34	6, 33	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
35	30, 31, 32, 34	qusbas 17464	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
36		nsgsubg 19085	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
37		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~_QG 𝐾) = (𝐺 ~_QG 𝐾)
38	23, 37	eqger 19105	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝐾) Er (Base‘𝐺))
39	10, 36, 38	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) Er (Base‘𝐺))
40	39	qsss 8711	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
41	35, 40	eqsstrrd 3967	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
42	41	sselda 3931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
43	42	elpwid 4561	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
44	43	sselda 3931	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) → 𝑥 ∈ (Base‘𝐺))
45	44	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
46	29, 45	fnfvelrnd 7025	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ ran 𝐹)
47	27, 46	sseldd 3932	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
48	22, 47	eqeltrd 2834	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
49	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
50		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
51	7, 49, 5, 11, 20, 50	ghmquskerlem2 19212	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
52	48, 51	r19.29a 3142	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
53	19, 21, 52	fmpt2d 7067	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶(Base‘𝐻))
54	39	ad6antr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐺 ~_QG 𝐾) Er (Base‘𝐺))
55	50	ad5antr 734	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ∈ (Base‘𝑄))
56	35	ad6antr 736	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
57	55, 56	eleqtrrd 2837	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝐾)))
58		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ 𝑟)
59		qsel 8731	. . . . . . . . . . 11 ⊢ (((𝐺 ~_QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) ∧ 𝑥 ∈ 𝑟) → 𝑟 = [𝑥](𝐺 ~_QG 𝐾))
60	54, 57, 58, 59	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 = [𝑥](𝐺 ~_QG 𝐾))
61		simp-5r 785	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ∈ (Base‘𝑄))
62	61, 56	eleqtrrd 2837	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝐾)))
63		simplr 768	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ 𝑠)
64		qsel 8731	. . . . . . . . . . 11 ⊢ (((𝐺 ~_QG 𝐾) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) ∧ 𝑦 ∈ 𝑠) → 𝑠 = [𝑦](𝐺 ~_QG 𝐾))
65	54, 62, 63, 64	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 = [𝑦](𝐺 ~_QG 𝐾))
66	60, 65	oveq12d 7374	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝑟(+_g‘𝑄)𝑠) = ([𝑥](𝐺 ~_QG 𝐾)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝐾)))
67	10	ad6antr 736	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐾 ∈ (NrmSGrp‘𝐺))
68	43	ad5antr 734	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ⊆ (Base‘𝐺))
69	68, 58	sseldd 3932	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ (Base‘𝐺))
70	41	sselda 3931	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
71	70	elpwid 4561	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
72	71	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
73	72	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ⊆ (Base‘𝐺))
74	73, 63	sseldd 3932	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ (Base‘𝐺))
75		eqid 2734	. . . . . . . . . . 11 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
76	11, 23, 75, 3	qusadd 19115	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ([𝑥](𝐺 ~_QG 𝐾)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝐾)) = [(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝐾))
77	67, 69, 74, 76	syl3anc 1373	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ([𝑥](𝐺 ~_QG 𝐾)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝐾)) = [(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝐾))
78	66, 77	eqtrd 2769	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝑟(+_g‘𝑄)𝑠) = [(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝐾))
79	78	fveq2d 6836	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = (𝐽‘[(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝐾)))
80	6	ad6antr 736	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
81	80, 33	syl 17	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐺 ∈ Grp)
82	23, 75, 81, 69, 74	grpcld 18875	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
83	7, 80, 5, 11, 20, 82	ghmquskerlem1 19210	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘[(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝐾)) = (𝐹‘(𝑥(+_g‘𝐺)𝑦)))
84	23, 75, 4	ghmlin 19148	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐹‘𝑥)(+_g‘𝐻)(𝐹‘𝑦)))
85	80, 69, 74, 84	syl3anc 1373	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐹‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐹‘𝑥)(+_g‘𝐻)(𝐹‘𝑦)))
86	79, 83, 85	3eqtrd 2773	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐹‘𝑥)(+_g‘𝐻)(𝐹‘𝑦)))
87		simpllr 775	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘𝑟) = (𝐹‘𝑥))
88		simpr 484	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘𝑠) = (𝐹‘𝑦))
89	87, 88	oveq12d 7374	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)) = ((𝐹‘𝑥)(+_g‘𝐻)(𝐹‘𝑦)))
90	86, 89	eqtr4d 2772	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
91	6	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
92		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑠 ∈ (Base‘𝑄))
93	7, 91, 5, 11, 20, 92	ghmquskerlem2 19212	. . . . 5 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ∃𝑦 ∈ 𝑠 (𝐽‘𝑠) = (𝐹‘𝑦))
94	90, 93	r19.29a 3142	. . . 4 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
95	51	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
96	94, 95	r19.29a 3142	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
97	96	anasss 466	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
98	1, 2, 3, 4, 13, 16, 53, 97	isghmd 19152	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∃wrex 3058 Vcvv 3438 ⊆ wss 3899 𝒫 cpw 4552 {csn 4578 ∪ cuni 4861 ↦ cmpt 5177 ^◡ccnv 5621 ran crn 5623 “ cima 5625 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 Er wer 8630 [cec 8631 / cqs 8632 Basecbs 17134 +_gcplusg 17175 0_gc0g 17357 /_s cqus 17424 Grpcgrp 18861 SubGrpcsubg 19048 NrmSGrpcnsg 19049 ~_QG cqg 19050 GrpHom cghm 19139

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 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 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-0g 17359 df-imas 17427 df-qus 17428 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-nsg 19052 df-eqg 19053 df-ghm 19140

This theorem is referenced by: ghmqusker 19214 rhmquskerlem 33455

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