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Theorem ghmquskerlem3 19218

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 2729	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2729	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2729	. 2 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
4		eqid 2729	. 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 19174	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
10	5, 9	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
11		ghmqusker.q	. . . 4 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾))
12	11	qusgrp 19118	. . 3 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
13	10, 12	syl 17	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
14		ghmrn 19161	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻))
15		subgrcl 19063	. . 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 7892	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
19	18	uniexd 7718	. . 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 2729	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
24	23, 2	ghmf 19152	. . . . . . . . 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 6696	. . . . . . 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 6689	. . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
32		ovexd 7422	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
33		ghmgrp1 19150	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
34	6, 33	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
35	30, 31, 32, 34	qusbas 17508	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
36		nsgsubg 19090	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
37		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~_QG 𝐾) = (𝐺 ~_QG 𝐾)
38	23, 37	eqger 19110	. . . . . . . . . . . . . 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 8749	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
41	35, 40	eqsstrrd 3982	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
42	41	sselda 3946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
43	42	elpwid 4572	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
44	43	sselda 3946	. . . . . . . 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 7054	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ ran 𝐹)
47	27, 46	sseldd 3947	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
48	22, 47	eqeltrd 2828	. . . 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 19217	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
52	48, 51	r19.29a 3141	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
53	19, 21, 52	fmpt2d 7096	. 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 2831	. . . . . . . . . . 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 8769	. . . . . . . . . . 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 2831	. . . . . . . . . . 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 8769	. . . . . . . . . . 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 7405	. . . . . . . . 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 3947	. . . . . . . . . 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 3946	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
71	70	elpwid 4572	. . . . . . . . . . . . 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 3947	. . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
76	11, 23, 75, 3	qusadd 19120	. . . . . . . . . 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 2764	. . . . . . . 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 6862	. . . . . . 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 18879	. . . . . . . 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 19215	. . . . . . 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 19153	. . . . . . . 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 2768	. . . . . 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 7405	. . . . . 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 2767	. . . . 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 19217	. . . . 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 3141	. . . 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 3141	. . 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 19157	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 𝒫 cpw 4563 {csn 4589 ∪ cuni 4871 ↦ cmpt 5188 ^◡ccnv 5637 ran crn 5639 “ cima 5641 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 Er wer 8668 [cec 8669 / cqs 8670 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 /_s cqus 17468 Grpcgrp 18865 SubGrpcsubg 19052 NrmSGrpcnsg 19053 ~_QG cqg 19054 GrpHom cghm 19144

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-0g 17404 df-imas 17471 df-qus 17472 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145

This theorem is referenced by: ghmqusker 19219 rhmquskerlem 33396

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