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Theorem ghmquskerlem3 19402

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 2765	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2765	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2765	. 2 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
4		eqid 2765	. 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 19358	. . . . 5 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
9	6, 8	syl 18	. . . 4 ⊢ (𝜑 → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
10	5, 9	eqeltrid 2869	. . 3 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
11		ghmqusker.q	. . . 4 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾))
12	11	qusgrp 19303	. . 3 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
13	10, 12	syl 18	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
14		ghmrn 19345	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻))
15		subgrcl 19243	. . 3 ⊢ (ran 𝐹 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
16	6, 14, 15	3syl 19	. 2 ⊢ (𝜑 → 𝐻 ∈ Grp)
17	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
18	17	imaexd 7919	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ (Base‘𝑄)) → (𝐹 “ 𝑞) ∈ V)
19	18	uniexd 7750	. . 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 490	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) = (𝐹‘𝑥))
23		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
24	23, 2	ghmf 19336	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
25	6, 24	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
26	25	frnd 6718	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐻))
27	26	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ran 𝐹 ⊆ (Base‘𝐻))
28	25	ffnd 6710	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
29	28	ad3antrrr 743	. . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
32		ovexd 7454	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
33		ghmgrp1 19334	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
34	6, 33	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Grp)
35	30, 31, 32, 34	qusbas 17623	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
36		nsgsubg 19270	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
37		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~_QG 𝐾) = (𝐺 ~_QG 𝐾)
38	23, 37	eqger 19292	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝐾) Er (Base‘𝐺))
39	10, 36, 38	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) Er (Base‘𝐺))
40	39	qsss 8779	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
41	35, 40	eqsstrrd 3973	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
42	41	sselda 3938	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
43	42	elpwid 4573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
44	43	sselda 3938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) → 𝑥 ∈ (Base‘𝐺))
45	44	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑥 ∈ (Base‘𝐺))
46	29, 45	fnfvelrnd 7081	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ ran 𝐹)
47	27, 46	sseldd 3939	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ (Base‘𝐻))
48	22, 47	eqeltrd 2865	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
49	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
50		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
51	7, 49, 5, 11, 20, 50	ghmquskerlem2 19401	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
52	48, 51	r19.29a 3175	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → (𝐽‘𝑟) ∈ (Base‘𝐻))
53	19, 21, 52	fmpt2d 7124	. 2 ⊢ (𝜑 → 𝐽:(Base‘𝑄)⟶(Base‘𝐻))
54	39	ad6antr 749	. . . . . . . . . . 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 747	. . . . . . . . . . . 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 749	. . . . . . . . . . . 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 2868	. . . . . . . . . . 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 796	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ 𝑟)
59		qsel 8800	. . . . . . . . . . 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 1398	. . . . . . . . . 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 798	. . . . . . . . . . . 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 2868	. . . . . . . . . . 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 781	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ 𝑠)
64		qsel 8800	. . . . . . . . . . 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 1398	. . . . . . . . . 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 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 𝐾)(+_g‘𝑄)[𝑦](𝐺 ~_QG 𝐾)))
67	10	ad6antr 749	. . . . . . . . . 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 747	. . . . . . . . . . 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 3939	. . . . . . . . . 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 3938	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
71	70	elpwid 4573	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
72	71	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
73	72	ad4antr 745	. . . . . . . . . . 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 3939	. . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
76	11, 23, 75, 3	qusadd 19305	. . . . . . . . . 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 1398	. . . . . . . . 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 2800	. . . . . . . 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 6889	. . . . . . 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 749	. . . . . . . 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 18	. . . . . . . . 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 19060	. . . . . . . 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 19399	. . . . . . 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 19337	. . . . . . . 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 1398	. . . . . . 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 2804	. . . . . 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 788	. . . . . . 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 490	. . . . . . 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 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)))
90	86, 89	eqtr4d 2803	. . . . 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 745	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
92		simpllr 788	. . . . . 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 19401	. . . . 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 3175	. . . 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 486	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
96	94, 95	r19.29a 3175	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(+_g‘𝑄)𝑠)) = ((𝐽‘𝑟)(+_g‘𝐻)(𝐽‘𝑠)))
97	96	anasss 472	. 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 19341	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 Vcvv 3457 ⊆ wss 3906 𝒫 cpw 4564 {csn 4591 ∪ cuni 4874 ↦ cmpt 5194 ^◡ccnv 5662 ran crn 5664 “ cima 5666 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 Er wer 8697 [cec 8698 / cqs 8699 Basecbs 17293 +_gcplusg 17334 0_gc0g 17516 /_s cqus 17583 Grpcgrp 19046 SubGrpcsubg 19232 NrmSGrpcnsg 19233 ~_QG cqg 19234 GrpHom cghm 19329

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-0g 17518 df-imas 17586 df-qus 17587 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-grp 19049 df-minusg 19050 df-sbg 19051 df-subg 19235 df-nsg 19236 df-eqg 19237 df-ghm 19330

This theorem is used by: ghmqusker 19403 rhmquskerlem 33799

W3C validator