ghmqusnsglem1 - Mathbox for Thierry Arnoux

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Theorem ghmqusnsglem1 33123

Description: Lemma for ghmqusnsg 33125. (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))
ghmqusnsglem1.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))

**Assertion**
Ref	Expression
ghmqusnsglem1	⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~_QG 𝑁)) = (𝐹‘𝑋))

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

Proof of Theorem ghmqusnsglem1 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmqusnsg.j	. . 3 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
2		imaeq2 6053	. . . 4 ⊢ (𝑞 = [𝑋](𝐺 ~_QG 𝑁) → (𝐹 “ 𝑞) = (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)))
3	2	unieqd 4916	. . 3 ⊢ (𝑞 = [𝑋](𝐺 ~_QG 𝑁) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)))
4		ghmqusnsglem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))
5		ovex 7447	. . . . . 6 ⊢ (𝐺 ~_QG 𝑁) ∈ V
6	5	ecelqsi 8785	. . . . 5 ⊢ (𝑋 ∈ (Base‘𝐺) → [𝑋](𝐺 ~_QG 𝑁) ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)))
7	4, 6	syl 17	. . . 4 ⊢ (𝜑 → [𝑋](𝐺 ~_QG 𝑁) ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)))
8		ghmqusnsg.q	. . . . . 6 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁))
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁)))
10		eqidd 2729	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
11		ovexd 7449	. . . . 5 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) ∈ V)
12		ghmqusnsg.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
13		ghmgrp1 19165	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
15	9, 10, 11, 14	qusbas 17520	. . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) = (Base‘𝑄))
16	7, 15	eleqtrd 2831	. . 3 ⊢ (𝜑 → [𝑋](𝐺 ~_QG 𝑁) ∈ (Base‘𝑄))
17	12	imaexd 7918	. . . 4 ⊢ (𝜑 → (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) ∈ V)
18	17	uniexd 7741	. . 3 ⊢ (𝜑 → ∪ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) ∈ V)
19	1, 3, 16, 18	fvmptd3 7022	. 2 ⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~_QG 𝑁)) = ∪ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)))
20		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
21		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘𝐻) = (Base‘𝐻)
22	20, 21	ghmf 19167	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
23	12, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
24	23	ffnd 6717	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐺))
25		ghmqusnsg.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
26		nsgsubg 19106	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
27		eqid 2728	. . . . . . . . . 10 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
28	20, 27	eqger 19126	. . . . . . . . 9 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
29	25, 26, 28	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
30	29	ecss 8765	. . . . . . 7 ⊢ (𝜑 → [𝑋](𝐺 ~_QG 𝑁) ⊆ (Base‘𝐺))
31	24, 30	fvelimabd 6966	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) ↔ ∃𝑧 ∈ [ 𝑋](𝐺 ~_QG 𝑁)(𝐹‘𝑧) = 𝑦))
32		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) ∧ (𝐹‘𝑧) = 𝑦) → (𝐹‘𝑧) = 𝑦)
33	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
34		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
35	33, 13	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝐺 ∈ Grp)
36	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝑋 ∈ (Base‘𝐺))
37	20, 34, 35, 36	grpinvcld 18938	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((inv_g‘𝐺)‘𝑋) ∈ (Base‘𝐺))
38	30	sselda 3978	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝑧 ∈ (Base‘𝐺))
39		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
40		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
41	20, 39, 40	ghmlin 19168	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ ((inv_g‘𝐺)‘𝑋) ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘(((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧)) = ((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧)))
42	33, 37, 38, 41	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (𝐹‘(((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧)) = ((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧)))
43	24	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝐹 Fn (Base‘𝐺))
44		ghmqusnsg.n	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
45		ghmqusnsg.k	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐾 = (^◡𝐹 “ { 0 })
46	44, 45	sseqtrdi 4028	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ⊆ (^◡𝐹 “ { 0 }))
47	46	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝑁 ⊆ (^◡𝐹 “ { 0 }))
48	20	subgss 19075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ (Base‘𝐺))
49	25, 26, 48	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ⊆ (Base‘𝐺))
50	49	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝑁 ⊆ (Base‘𝐺))
51		vex 3474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
52		elecg 8761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ V ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁) ↔ 𝑋(𝐺 ~_QG 𝑁)𝑧))
53	51, 52	mpan 689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ (Base‘𝐺) → (𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁) ↔ 𝑋(𝐺 ~_QG 𝑁)𝑧))
54	53	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝑋(𝐺 ~_QG 𝑁)𝑧)
55	4, 54	sylan 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝑋(𝐺 ~_QG 𝑁)𝑧)
56	20, 34, 39, 27	eqgval 19125	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ⊆ (Base‘𝐺)) → (𝑋(𝐺 ~_QG 𝑁)𝑧 ↔ (𝑋 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ (((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ 𝑁)))
57	56	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ (Base‘𝐺)) ∧ 𝑋(𝐺 ~_QG 𝑁)𝑧) → (𝑋 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ (((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ 𝑁))
58	57	simp3d 1142	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ (Base‘𝐺)) ∧ 𝑋(𝐺 ~_QG 𝑁)𝑧) → (((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ 𝑁)
59	35, 50, 55, 58	syl21anc 837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ 𝑁)
60	47, 59	sseldd 3979	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ (^◡𝐹 “ { 0 }))
61		fniniseg 7063	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn (Base‘𝐺) → ((((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ (^◡𝐹 “ { 0 }) ↔ ((((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ (Base‘𝐺) ∧ (𝐹‘(((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧)) = 0 )))
62	61	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn (Base‘𝐺) ∧ (((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ (^◡𝐹 “ { 0 })) → ((((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ (Base‘𝐺) ∧ (𝐹‘(((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧)) = 0 ))
63	43, 60, 62	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧) ∈ (Base‘𝐺) ∧ (𝐹‘(((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧)) = 0 ))
64	63	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (𝐹‘(((inv_g‘𝐺)‘𝑋)(+_g‘𝐺)𝑧)) = 0 )
65	42, 64	eqtr3d 2770	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧)) = 0 )
66	65	oveq2d 7430	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘𝑋)(+_g‘𝐻)((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧))) = ((𝐹‘𝑋)(+_g‘𝐻) 0 ))
67		eqid 2728	. . . . . . . . . . . . . . . . 17 ⊢ (inv_g‘𝐻) = (inv_g‘𝐻)
68	20, 34, 67	ghminv 19170	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑋 ∈ (Base‘𝐺)) → (𝐹‘((inv_g‘𝐺)‘𝑋)) = ((inv_g‘𝐻)‘(𝐹‘𝑋)))
69	33, 36, 68	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (𝐹‘((inv_g‘𝐺)‘𝑋)) = ((inv_g‘𝐻)‘(𝐹‘𝑋)))
70	69	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧)) = (((inv_g‘𝐻)‘(𝐹‘𝑋))(+_g‘𝐻)(𝐹‘𝑧)))
71	70	oveq2d 7430	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘𝑋)(+_g‘𝐻)((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧))) = ((𝐹‘𝑋)(+_g‘𝐻)(((inv_g‘𝐻)‘(𝐹‘𝑋))(+_g‘𝐻)(𝐹‘𝑧))))
72		ghmgrp2 19166	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
73	33, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝐻 ∈ Grp)
74	33, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
75	74, 36	ffvelcdmd 7089	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (𝐹‘𝑋) ∈ (Base‘𝐻))
76	74, 38	ffvelcdmd 7089	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
77	21, 40, 67	grpasscan1 18951	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ Grp ∧ (𝐹‘𝑋) ∈ (Base‘𝐻) ∧ (𝐹‘𝑧) ∈ (Base‘𝐻)) → ((𝐹‘𝑋)(+_g‘𝐻)(((inv_g‘𝐻)‘(𝐹‘𝑋))(+_g‘𝐻)(𝐹‘𝑧))) = (𝐹‘𝑧))
78	73, 75, 76, 77	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘𝑋)(+_g‘𝐻)(((inv_g‘𝐻)‘(𝐹‘𝑋))(+_g‘𝐻)(𝐹‘𝑧))) = (𝐹‘𝑧))
79	71, 78	eqtrd 2768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘𝑋)(+_g‘𝐻)((𝐹‘((inv_g‘𝐺)‘𝑋))(+_g‘𝐻)(𝐹‘𝑧))) = (𝐹‘𝑧))
80		ghmqusnsg.0	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝐻)
81	21, 40, 80, 73, 75	grpridd 18920	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → ((𝐹‘𝑋)(+_g‘𝐻) 0 ) = (𝐹‘𝑋))
82	66, 79, 81	3eqtr3d 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) → (𝐹‘𝑧) = (𝐹‘𝑋))
83	82	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) ∧ (𝐹‘𝑧) = 𝑦) → (𝐹‘𝑧) = (𝐹‘𝑋))
84	32, 83	eqtr3d 2770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ [𝑋](𝐺 ~_QG 𝑁)) ∧ (𝐹‘𝑧) = 𝑦) → 𝑦 = (𝐹‘𝑋))
85	84	r19.29an 3154	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑧 ∈ [ 𝑋](𝐺 ~_QG 𝑁)(𝐹‘𝑧) = 𝑦) → 𝑦 = (𝐹‘𝑋))
86		fveqeq2 6900	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = 𝑦 ↔ (𝐹‘𝑋) = 𝑦))
87		ecref 8762	. . . . . . . . . . 11 ⊢ (((𝐺 ~_QG 𝑁) Er (Base‘𝐺) ∧ 𝑋 ∈ (Base‘𝐺)) → 𝑋 ∈ [𝑋](𝐺 ~_QG 𝑁))
88	29, 4, 87	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ [𝑋](𝐺 ~_QG 𝑁))
89	88	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑋 ∈ [𝑋](𝐺 ~_QG 𝑁))
90		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑦 = (𝐹‘𝑋))
91	90	eqcomd 2734	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐹‘𝑋) = 𝑦)
92	86, 89, 91	rspcedvdw 3611	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ∃𝑧 ∈ [ 𝑋](𝐺 ~_QG 𝑁)(𝐹‘𝑧) = 𝑦)
93	85, 92	impbida 800	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ [ 𝑋](𝐺 ~_QG 𝑁)(𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑋)))
94		velsn 4640	. . . . . . 7 ⊢ (𝑦 ∈ {(𝐹‘𝑋)} ↔ 𝑦 = (𝐹‘𝑋))
95	93, 94	bitr4di 289	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ [ 𝑋](𝐺 ~_QG 𝑁)(𝐹‘𝑧) = 𝑦 ↔ 𝑦 ∈ {(𝐹‘𝑋)}))
96	31, 95	bitrd 279	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) ↔ 𝑦 ∈ {(𝐹‘𝑋)}))
97	96	eqrdv 2726	. . . 4 ⊢ (𝜑 → (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) = {(𝐹‘𝑋)})
98	97	unieqd 4916	. . 3 ⊢ (𝜑 → ∪ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) = ∪ {(𝐹‘𝑋)})
99		fvex 6904	. . . 4 ⊢ (𝐹‘𝑋) ∈ V
100	99	unisn 4924	. . 3 ⊢ ∪ {(𝐹‘𝑋)} = (𝐹‘𝑋)
101	98, 100	eqtrdi 2784	. 2 ⊢ (𝜑 → ∪ (𝐹 “ [𝑋](𝐺 ~_QG 𝑁)) = (𝐹‘𝑋))
102	19, 101	eqtrd 2768	1 ⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~_QG 𝑁)) = (𝐹‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 Vcvv 3470 ⊆ wss 3945 {csn 4624 ∪ cuni 4903 class class class wbr 5142 ↦ cmpt 5225 ^◡ccnv 5671 “ cima 5675 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Er wer 8715 [cec 8716 / cqs 8717 Basecbs 17173 +_gcplusg 17226 0_gc0g 17414 /_s cqus 17480 Grpcgrp 18883 inv_gcminusg 18884 SubGrpcsubg 19068 NrmSGrpcnsg 19069 ~_QG cqg 19070 GrpHom cghm 19160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-0g 17416 df-imas 17483 df-qus 17484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-subg 19071 df-nsg 19072 df-eqg 19073 df-ghm 19161

This theorem is referenced by: ghmqusnsglem2 33124 ghmqusnsg 33125 rhmqusnsg 33137

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