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Theorem ghmqusnsg 19245

Description: The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over a normal subgroup 𝑁 of 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, 13-May-2025.)

**Hypotheses**
Ref	Expression
ghmqusnsg.0	⊢ 0 = (0_g‘𝐻)
ghmqusnsg.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusnsg.k	⊢ 𝐾 = (^◡𝐹 “ { 0 })
ghmqusnsg.q	⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁))
ghmqusnsg.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
ghmqusnsg.n	⊢ (𝜑 → 𝑁 ⊆ 𝐾)
ghmqusnsg.1	⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))

**Assertion**
Ref	Expression
ghmqusnsg	⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))

Distinct variable groups: 𝐹,𝑞 𝐺,𝑞 𝐾,𝑞 𝑁,𝑞 𝑄,𝑞 𝜑,𝑞 𝐻,𝑞 𝐽,𝑞 Allowed substitution hint: 0 (𝑞)

Proof of Theorem ghmqusnsg Dummy variables 𝑦 𝑥 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2725	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2725	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2725	. 2 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
4		eqid 2725	. 2 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
5		ghmqusnsg.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
6		ghmqusnsg.q	. . . 4 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁))
7	6	qusgrp 19149	. . 3 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
8	5, 7	syl 17	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
9		ghmqusnsg.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
10		ghmrn 19192	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻))
11		subgrcl 19094	. . 3 ⊢ (ran 𝐹 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
12	9, 10, 11	3syl 18	. 2 ⊢ (𝜑 → 𝐻 ∈ Grp)
13	9	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
14	13	imaexd 7924	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
15	14	uniexd 7748	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → ∪ (𝐹 “ 𝑞) ∈ V)
16		ghmqusnsg.j	. . . 4 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)))
18		simpr 483	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
19		eqid 2725	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
20	19, 2	ghmf 19183	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
21	9, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
22	21	frnd 6731	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐻))
23	22	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ran 𝐹 ⊆ (Base‘𝐻))
24	21	ffnd 6724	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
25	24	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 Fn (Base‘𝐺))
26	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁)))
27		eqidd 2726	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
28		ovexd 7454	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) ∈ V)
29		ghmgrp1 19181	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
30	9, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
31	26, 27, 28, 30	qusbas 17530	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) = (Base‘𝑄))
32		nsgsubg 19121	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
33		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
34	19, 33	eqger 19141	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
35	5, 32, 34	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
36	35	qsss 8797	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) ⊆ 𝒫 (Base‘𝐺))
37	31, 36	eqsstrrd 4016	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
38	37	sselda 3976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
39	38	elpwid 4613	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
40	39	sselda 3976	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) → 𝑥 ∈ (Base‘𝐺))
41	40	adantr 479	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
42	25, 41	fnfvelrnd 7091	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ ran 𝐹)
43	23, 42	sseldd 3977	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
44	18, 43	eqeltrd 2825	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
45		ghmqusnsg.0	. . . . 5 ⊢ 0 = (0_g‘𝐻)
46	9	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
47		ghmqusnsg.k	. . . . 5 ⊢ 𝐾 = (^◡𝐹 “ { 0 })
48		ghmqusnsg.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
49	48	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑁 ⊆ 𝐾)
50	5	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
51		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
52	45, 46, 47, 6, 16, 49, 50, 51	ghmqusnsglem2 19244	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
53	44, 52	r19.29a 3151	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
54	15, 17, 53	fmpt2d 7133	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶(Base‘𝐻))
55	35	ad6antr 734	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
56	51	ad5antr 732	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ∈ (Base‘𝑄))
57	31	ad6antr 734	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) = (Base‘𝑄))
58	56, 57	eleqtrrd 2828	. . . . . . . . . . 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 𝑁)))
59		simp-4r 782	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ 𝑟)
60		qsel 8815	. . . . . . . . . . 11 ⊢ (((𝐺 ~_QG 𝑁) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) ∧ 𝑥 ∈ 𝑟) → 𝑟 = [𝑥](𝐺 ~_QG 𝑁))
61	55, 58, 59, 60	syl3anc 1368	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 = [𝑥](𝐺 ~_QG 𝑁))
62		simp-5r 784	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ∈ (Base‘𝑄))
63	62, 57	eleqtrrd 2828	. . . . . . . . . . 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 𝑁)))
64		simplr 767	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ 𝑠)
65		qsel 8815	. . . . . . . . . . 11 ⊢ (((𝐺 ~_QG 𝑁) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) ∧ 𝑦 ∈ 𝑠) → 𝑠 = [𝑦](𝐺 ~_QG 𝑁))
66	55, 63, 64, 65	syl3anc 1368	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 = [𝑦](𝐺 ~_QG 𝑁))
67	61, 66	oveq12d 7437	. . . . . . . . 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 𝑁)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝑁)))
68	5	ad6antr 734	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑁 ∈ (NrmSGrp‘𝐺))
69	39	ad5antr 732	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ⊆ (Base‘𝐺))
70	69, 59	sseldd 3977	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ (Base‘𝐺))
71	37	sselda 3976	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
72	71	elpwid 4613	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
73	72	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
74	73	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ⊆ (Base‘𝐺))
75	74, 64	sseldd 3977	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ (Base‘𝐺))
76		eqid 2725	. . . . . . . . . . 11 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
77	6, 19, 76, 3	qusadd 19151	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ([𝑥](𝐺 ~_QG 𝑁)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝑁)) = [(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝑁))
78	68, 70, 75, 77	syl3anc 1368	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ([𝑥](𝐺 ~_QG 𝑁)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝑁)) = [(𝑥(+_g‘𝐺)𝑦)](𝐺 ~_QG 𝑁))
79	67, 78	eqtrd 2765	. . . . . . . 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 𝑁))
80	79	fveq2d 6900	. . . . . . 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 𝑁)))
81	9	ad6antr 734	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
82	48	ad6antr 734	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑁 ⊆ 𝐾)
83	81, 29	syl 17	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐺 ∈ Grp)
84	19, 76, 83, 70, 75	grpcld 18912	. . . . . . . 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))
85	45, 81, 47, 6, 16, 82, 68, 84	ghmqusnsglem1 19243	. . . . . . 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 𝑁)) = (𝐹‘(𝑥(+_g‘𝐺)𝑦)))
86	19, 76, 4	ghmlin 19184	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐹‘𝑥)(+_g‘𝐻)(𝐹‘𝑦)))
87	81, 70, 75, 86	syl3anc 1368	. . . . . . 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)))
88	80, 85, 87	3eqtrd 2769	. . . . . 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)))
89		simpllr 774	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘𝑟) = (𝐹‘𝑥))
90		simpr 483	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘𝑠) = (𝐹‘𝑦))
91	89, 90	oveq12d 7437	. . . . . 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)))
92	88, 91	eqtr4d 2768	. . . . 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)))
93	9	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
94	48	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑁 ⊆ 𝐾)
95	5	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑁 ∈ (NrmSGrp‘𝐺))
96		simpllr 774	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑠 ∈ (Base‘𝑄))
97	45, 93, 47, 6, 16, 94, 95, 96	ghmqusnsglem2 19244	. . . . 5 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ∃𝑦 ∈ 𝑠 (𝐽‘𝑠) = (𝐹‘𝑦))
98	92, 97	r19.29a 3151	. . . 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)))
99	52	adantr 479	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
100	98, 99	r19.29a 3151	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
101	100	anasss 465	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
102	1, 2, 3, 4, 8, 12, 54, 101	isghmd 19188	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 Vcvv 3461 ⊆ wss 3944 𝒫 cpw 4604 {csn 4630 ∪ cuni 4909 ↦ cmpt 5232 ^◡ccnv 5677 ran crn 5679 “ cima 5681 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Er wer 8722 [cec 8723 / cqs 8724 Basecbs 17183 +_gcplusg 17236 0_gc0g 17424 /_s cqus 17490 Grpcgrp 18898 SubGrpcsubg 19083 NrmSGrpcnsg 19084 ~_QG cqg 19085 GrpHom cghm 19175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-0g 17426 df-imas 17493 df-qus 17494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-subg 19086 df-nsg 19087 df-eqg 19088 df-ghm 19176

This theorem is referenced by: rhmqusnsg 21192

W3C validator