Theorem nsgqusf1olem1 31131

Description: Lemma for nsgqusf1o 31134. (Contributed by Thierry Arnoux, 4-Aug-2024.)

Hypotheses

Ref	Expression
nsgqusf1o.b	⊢ 𝐵 = (Base‘𝐺)
nsgqusf1o.s	⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}
nsgqusf1o.t	⊢ 𝑇 = (SubGrp‘𝑄)
nsgqusf1o.1	⊢ ≤ = (le‘(toInc‘𝑆))
nsgqusf1o.2	⊢ ≲ = (le‘(toInc‘𝑇))
nsgqusf1o.q	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
nsgqusf1o.p	⊢ ⊕ = (LSSum‘𝐺)
nsgqusf1o.e	⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
nsgqusf1o.f	⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
nsgqusf1o.n	⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))

Assertion

Ref	Expression
nsgqusf1olem1	⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)

Distinct variable groups:   ⊕ ,𝑎,𝑓,ℎ,𝑥   𝐵,𝑎,𝑓,ℎ,𝑥   𝐸,𝑎,𝑓,ℎ,𝑥   𝑓,𝐹,ℎ,𝑥   𝐺,𝑎,𝑓,ℎ,𝑥   𝑁,𝑎,𝑓,ℎ,𝑥   𝑄,𝑎,𝑓,ℎ,𝑥   𝑆,𝑎,𝑓,ℎ,𝑥   𝑇,𝑎,𝑓,ℎ,𝑥   𝜑,𝑎,𝑓,ℎ,𝑥

Allowed substitution hints:   𝐹(𝑎)   ≤ (𝑥,𝑓,ℎ,𝑎)   ≲ (𝑥,𝑓,ℎ,𝑎)

Proof of Theorem nsgqusf1olem1

Dummy variables 𝑖 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nsgqusf1o.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
2		nsgqusf1o.q	. . . . . 6 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
3	2	qusgrp 18415	. . . . 5 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Grp)
5	4	ad2antrr 725	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → 𝑄 ∈ Grp)
6		nsgqusf1o.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
7	6	subgss 18360	. . . . . . . . 9 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵)
8	7	ad2antlr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ ⊆ 𝐵)
9	8	sselda 3894	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ 𝐵)
10		ovex 7189	. . . . . . . 8 ⊢ (𝐺 ~QG 𝑁) ∈ V
11	10	ecelqsi 8369	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
12	9, 11	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
13		nsgqusf1o.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
14		nsgsubg 18390	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
15	1, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
16	15	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → 𝑁 ∈ (SubGrp‘𝐺))
17	6, 13, 16, 9	quslsm 31126	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
18	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
19	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
20		ovexd 7191	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
21		subgrcl 18364	. . . . . . . . 9 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
22	15, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
23	18, 19, 20, 22	qusbas 16889	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
24	23	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
25	12, 17, 24	3eltr3d 2866	. . . . 5 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄))
26	25	ralrimiva 3113	. . . 4 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄))
27		eqid 2758	. . . . 5 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
28	27	rnmptss 6883	. . . 4 ⊢ (∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄))
29	26, 28	syl 17	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄))
30		nfv 1915	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ)
31		ovexd 7191	. . . 4 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ V)
32		eqid 2758	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
33	32	subg0cl 18367	. . . . . 6 ⊢ (ℎ ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ ℎ)
34	33	ne0d 4236	. . . . 5 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ≠ ∅)
35	34	ad2antlr 726	. . . 4 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ ≠ ∅)
36	30, 31, 27, 35	rnmptn0 6078	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅)
37		nfmpt1 5134	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
38	37	nfrn 5798	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
39	38	nfel2 2937	. . . . . 6 ⊢ Ⅎ𝑥 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
40	30, 39	nfan 1900	. . . . 5 ⊢ Ⅎ𝑥(((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
41	38	nfel2 2937	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
42	38, 41	nfralw 3153	. . . . . 6 ⊢ Ⅎ𝑥∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
43	38	nfel2 2937	. . . . . 6 ⊢ Ⅎ𝑥((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
44	42, 43	nfan 1900	. . . . 5 ⊢ Ⅎ𝑥(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
45		sneq 4535	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
46	45	oveq1d 7171	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ({𝑥} ⊕ 𝑁) = ({𝑧} ⊕ 𝑁))
47	46	cbvmptv 5139	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑧 ∈ ℎ ↦ ({𝑧} ⊕ 𝑁))
48		simp-4r 783	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
49	48	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
50		simp-4r 783	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑥 ∈ ℎ)
51		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑦 ∈ ℎ)
52		eqid 2758	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	52	subgcl 18369	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ ∧ 𝑦 ∈ ℎ) → (𝑥(+g‘𝐺)𝑦) ∈ ℎ)
54	49, 50, 51, 53	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑥(+g‘𝐺)𝑦) ∈ ℎ)
55		sneq 4535	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → {𝑧} = {(𝑥(+g‘𝐺)𝑦)})
56	55	oveq1d 7171	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → ({𝑧} ⊕ 𝑁) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁))
57	56	eqeq2d 2769	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → ((𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁) ↔ (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁)))
58	57	adantl 485	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) ∧ 𝑧 = (𝑥(+g‘𝐺)𝑦)) → ((𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁) ↔ (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁)))
59		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁))
60	17	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
61	59, 60	eqtr4d 2796	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
62	61	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
63		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑗 = ({𝑦} ⊕ 𝑁))
64	1	ad4antr 731	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
65	64	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
66	65, 14	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
67	49, 7	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ℎ ⊆ 𝐵)
68	67, 51	sseldd 3895	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑦 ∈ 𝐵)
69	6, 13, 66, 68	quslsm 31126	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → [𝑦](𝐺 ~QG 𝑁) = ({𝑦} ⊕ 𝑁))
70	63, 69	eqtr4d 2796	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑗 = [𝑦](𝐺 ~QG 𝑁))
71	62, 70	oveq12d 7174	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) = ([𝑥](𝐺 ~QG 𝑁)(+g‘𝑄)[𝑦](𝐺 ~QG 𝑁)))
72	9	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ 𝐵)
73	72	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑥 ∈ 𝐵)
74		eqid 2758	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑄) = (+g‘𝑄)
75	2, 6, 52, 74	qusadd 18417	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ([𝑥](𝐺 ~QG 𝑁)(+g‘𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g‘𝐺)𝑦)](𝐺 ~QG 𝑁))
76	65, 73, 68, 75	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ([𝑥](𝐺 ~QG 𝑁)(+g‘𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g‘𝐺)𝑦)](𝐺 ~QG 𝑁))
77	67, 54	sseldd 3895	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
78	6, 13, 66, 77	quslsm 31126	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → [(𝑥(+g‘𝐺)𝑦)](𝐺 ~QG 𝑁) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁))
79	71, 76, 78	3eqtrd 2797	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁))
80	54, 58, 79	rspcedvd 3546	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ∃𝑧 ∈ ℎ (𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁))
81		ovexd 7191	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) ∈ V)
82	47, 80, 81	elrnmptd 5807	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
83	82	adantllr 718	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
84		sneq 4535	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
85	84	oveq1d 7171	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ({𝑥} ⊕ 𝑁) = ({𝑦} ⊕ 𝑁))
86	85	cbvmptv 5139	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑦 ∈ ℎ ↦ ({𝑦} ⊕ 𝑁))
87		ovex 7189	. . . . . . . . . . . 12 ⊢ ({𝑦} ⊕ 𝑁) ∈ V
88	86, 87	elrnmpti 5806	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁))
89	88	biimpi 219	. . . . . . . . . 10 ⊢ (𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁))
90	89	adantl 485	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁))
91	83, 90	r19.29a 3213	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → (𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
92	91	ralrimiva 3113	. . . . . . 7 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
93		eqid 2758	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
94	93	subginvcl 18368	. . . . . . . . . 10 ⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ ℎ)
95	94	ad5ant24 760	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝐺)‘𝑥) ∈ ℎ)
96		simpr 488	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → 𝑦 = ((invg‘𝐺)‘𝑥))
97	96	sneqd 4537	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → {𝑦} = {((invg‘𝐺)‘𝑥)})
98	97	oveq1d 7171	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → ({𝑦} ⊕ 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁))
99	8	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ℎ ⊆ 𝐵)
100	94	ad4ant24 753	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ ℎ)
101	99, 100	sseldd 3895	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
102	6, 13, 16, 101	quslsm 31126	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁))
103	102	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁))
104	98, 103	eqtr4d 2796	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → ({𝑦} ⊕ 𝑁) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
105	104	eqeq2d 2769	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → (((invg‘𝑄)‘𝑖) = ({𝑦} ⊕ 𝑁) ↔ ((invg‘𝑄)‘𝑖) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁)))
106	61	fveq2d 6667	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) = ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)))
107		eqid 2758	. . . . . . . . . . . 12 ⊢ (invg‘𝑄) = (invg‘𝑄)
108	2, 6, 93, 107	qusinv 18419	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
109	64, 72, 108	syl2anc 587	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
110	106, 109	eqtrd 2793	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
111	95, 105, 110	rspcedvd 3546	. . . . . . . 8 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ∃𝑦 ∈ ℎ ((invg‘𝑄)‘𝑖) = ({𝑦} ⊕ 𝑁))
112		fvexd 6678	. . . . . . . 8 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) ∈ V)
113	86, 111, 112	elrnmptd 5807	. . . . . . 7 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
114	92, 113	jca 515	. . . . . 6 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
115	114	adantllr 718	. . . . 5 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
116		ovex 7189	. . . . . . . 8 ⊢ ({𝑥} ⊕ 𝑁) ∈ V
117	27, 116	elrnmpti 5806	. . . . . . 7 ⊢ (𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁))
118	117	biimpi 219	. . . . . 6 ⊢ (𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁))
119	118	adantl 485	. . . . 5 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁))
120	40, 44, 115, 119	r19.29af2 3253	. . . 4 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → (∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
121	120	ralrimiva 3113	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∀𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
122		eqid 2758	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
123	122, 74, 107	issubg2 18374	. . . 4 ⊢ (𝑄 ∈ Grp → (ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄) ↔ (ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))))
124	123	biimpar 481	. . 3 ⊢ ((𝑄 ∈ Grp ∧ (ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∧ ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
125	5, 29, 36, 121, 124	syl13anc 1369	. 2 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
126		nsgqusf1o.t	. 2 ⊢ 𝑇 = (SubGrp‘𝑄)
127	125, 126	eleqtrrdi 2863	1 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  {csn 4525   ↦ cmpt 5116  ran crn 5529  ‘cfv 6340  (class class class)co 7156  [cec 8303   / cqs 8304  Basecbs 16554  +_gcplusg 16636  lecple 16643  0_gc0g 16784   /_s cqus 16849  toInccipo 17840  Grpcgrp 18182  inv_gcminusg 18183  SubGrpcsubg 18353  NrmSGrpcnsg 18354   ~_QG cqg 18355  LSSumclsm 18839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-tpos 7908  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-ec 8307  df-qs 8311  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-sup 8952  df-inf 8953  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-struct 16556  df-ndx 16557  df-slot 16558  df-base 16560  df-sets 16561  df-ress 16562  df-plusg 16649  df-mulr 16650  df-sca 16652  df-vsca 16653  df-ip 16654  df-tset 16655  df-ple 16656  df-ds 16658  df-0g 16786  df-imas 16852  df-qus 16853  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-grp 18185  df-minusg 18186  df-subg 18356  df-nsg 18357  df-eqg 18358  df-oppg 18554  df-lsm 18841

This theorem is referenced by:  nsgqusf1olem2  31132  nsgqusf1olem3  31133