Theorem nsgqusf1olem1 33872

Description: Lemma for nsgqusf1o 33875. (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 19337	. . . . 5 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
4	1, 3	syl 18	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Grp)
5	4	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → 𝑄 ∈ Grp)
6		nsgqusf1o.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
7	6	subgss 19273	. . . . . . . . 9 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵)
8	7	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ ⊆ 𝐵)
9	8	sselda 3931	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ 𝐵)
10		ovex 7448	. . . . . . . 8 ⊢ (𝐺 ~QG 𝑁) ∈ V
11	10	ecelqsi 8773	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
12	9, 11	syl 18	. . . . . 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 19304	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
15	1, 14	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
16	15	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → 𝑁 ∈ (SubGrp‘𝐺))
17	6, 13, 16, 9	quslsm 33864	. . . . . 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 7450	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
21		subgrcl 19277	. . . . . . . . 9 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
22	15, 21	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
23	18, 19, 20, 22	qusbas 17653	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
24	23	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
25	12, 17, 24	3eltr3d 2874	. . . . 5 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄))
26	25	ralrimiva 3154	. . . 4 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄))
27		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
28	27	rnmptss 7118	. . . 4 ⊢ (∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ (Base‘𝑄) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄))
29	26, 28	syl 18	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ (Base‘𝑄))
30		nfv 1947	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ)
31		ovexd 7450	. . . 4 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ V)
32		eqid 2760	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
33	32	subg0cl 19280	. . . . . 6 ⊢ (ℎ ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ ℎ)
34	33	ne0d 4288	. . . . 5 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ≠ ∅)
35	34	ad2antlr 740	. . . 4 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ ≠ ∅)
36	30, 31, 27, 35	rnmptn0 6241	. . 3 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ≠ ∅)
37		nfmpt1 5204	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
38	37	nfrn 5937	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
39	38	nfel2 2940	. . . . . 6 ⊢ Ⅎ𝑥 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
40	30, 39	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
41	38	nfel2 2940	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
42	38, 41	nfralw 3309	. . . . . 6 ⊢ Ⅎ𝑥∀𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))(𝑖(+g‘𝑄)𝑗) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
43	38	nfel2 2940	. . . . . 6 ⊢ Ⅎ𝑥((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
44	42, 43	nfan 1932	. . . . 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 4594	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
46	45	oveq1d 7430	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ({𝑥} ⊕ 𝑁) = ({𝑧} ⊕ 𝑁))
47	46	cbvmptv 5209	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑧 ∈ ℎ ↦ ({𝑧} ⊕ 𝑁))
48		simp-4r 796	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
49	48	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
50		simp-4r 796	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑥 ∈ ℎ)
51		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑦 ∈ ℎ)
52		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
53	52	subgcl 19282	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ ∧ 𝑦 ∈ ℎ) → (𝑥(+g‘𝐺)𝑦) ∈ ℎ)
54	49, 50, 51, 53	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → (𝑥(+g‘𝐺)𝑦) ∈ ℎ)
55		sneq 4594	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → {𝑧} = {(𝑥(+g‘𝐺)𝑦)})
56	55	oveq1d 7430	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → ({𝑧} ⊕ 𝑁) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁))
57	56	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑥(+g‘𝐺)𝑦) → ((𝑖(+g‘𝑄)𝑗) = ({𝑧} ⊕ 𝑁) ↔ (𝑖(+g‘𝑄)𝑗) = ({(𝑥(+g‘𝐺)𝑦)} ⊕ 𝑁)))
58	57	adantl 487	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁))
60	17	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
61	59, 60	eqtr4d 2798	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
62	61	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
63		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑗 = ({𝑦} ⊕ 𝑁))
64	1	ad4antr 745	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
65	64	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
66	65, 14	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
67	49, 7	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → ℎ ⊆ 𝐵)
68	67, 51	sseldd 3932	. . . . . . . . . . . . . . . 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 33864	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ 𝐵)
73	72	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 ∈ ℎ) ∧ 𝑗 = ({𝑦} ⊕ 𝑁)) → 𝑥 ∈ 𝐵)
74		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑄) = (+g‘𝑄)
75	2, 6, 52, 74	qusadd 19339	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 33864	. . . . . . . . . . . . 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 2799	. . . . . . . . . . . 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 3578	. . . . . . . . . . 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 7450	. . . . . . . . . . 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 5948	. . . . . . . . . 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 732	. . . . . . . . 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 4594	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
85	84	oveq1d 7430	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ({𝑥} ⊕ 𝑁) = ({𝑦} ⊕ 𝑁))
86	85	cbvmptv 5209	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑦 ∈ ℎ ↦ ({𝑦} ⊕ 𝑁))
87		ovex 7448	. . . . . . . . . . 11 ⊢ ({𝑦} ⊕ 𝑁) ∈ V
88	86, 87	elrnmpti 5947	. . . . . . . . . 10 ⊢ (𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁))
89	88	bilani 510	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑦 ∈ ℎ 𝑗 = ({𝑦} ⊕ 𝑁))
90	83, 89	r19.29a 3170	. . . . . . . 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)))
91	90	ralrimiva 3154	. . . . . . 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)))
92		eqid 2760	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
93	92	subginvcl 19281	. . . . . . . . . 10 ⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ ℎ)
94	93	ad5ant24 773	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝐺)‘𝑥) ∈ ℎ)
95		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → 𝑦 = ((invg‘𝐺)‘𝑥))
96	95	sneqd 4596	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → {𝑦} = {((invg‘𝐺)‘𝑥)})
97	96	oveq1d 7430	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → ({𝑦} ⊕ 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁))
98	8	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ℎ ⊆ 𝐵)
99	93	ad4ant24 767	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ ℎ)
100	98, 99	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
101	6, 13, 16, 100	quslsm 33864	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) → [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg‘𝐺)‘𝑥)} ⊕ 𝑁))
102	101	ad2antrr 739	. . . . . . . . . . 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))
103	97, 102	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → ({𝑦} ⊕ 𝑁) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
104	103	eqeq2d 2771	. . . . . . . . 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)))
105	61	fveq2d 6884	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) = ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)))
106		eqid 2760	. . . . . . . . . . . 12 ⊢ (invg‘𝑄) = (invg‘𝑄)
107	2, 6, 92, 106	qusinv 19341	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝐵) → ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
108	64, 72, 107	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
109	105, 108	eqtrd 2795	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) = [((invg‘𝐺)‘𝑥)](𝐺 ~QG 𝑁))
110	94, 104, 109	rspcedvd 3578	. . . . . . . 8 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ∃𝑦 ∈ ℎ ((invg‘𝑄)‘𝑖) = ({𝑦} ⊕ 𝑁))
111		fvexd 6895	. . . . . . . 8 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) ∈ V)
112	86, 110, 111	elrnmptd 5948	. . . . . . 7 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑥 ∈ ℎ) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → ((invg‘𝑄)‘𝑖) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
113	91, 112	jca 521	. . . . . 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))))
114	113	adantllr 732	. . . . 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))))
115		ovex 7448	. . . . . . 7 ⊢ ({𝑥} ⊕ 𝑁) ∈ V
116	27, 115	elrnmpti 5947	. . . . . 6 ⊢ (𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁))
117	116	bilani 510	. . . . 5 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑖 ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑥 ∈ ℎ 𝑖 = ({𝑥} ⊕ 𝑁))
118	40, 44, 114, 117	r19.29af2 3270	. . . 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))))
119	118	ralrimiva 3154	. . 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))))
120		eqid 2760	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
121	120, 74, 106	issubg2 19288	. . . 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))))))
122	121	biimpar 483	. . 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))
123	5, 29, 36, 119, 122	syl13anc 1399	. 2 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
124		nsgqusf1o.t	. 2 ⊢ 𝑇 = (SubGrp‘𝑄)
125	123, 124	eleqtrrdi 2871	1 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  {csn 4584   ↦ cmpt 5186  ran crn 5656  ‘cfv 6534  (class class class)co 7415  [cec 8698   / cqs 8699  Basecbs 17323  +_gcplusg 17364  lecple 17371  0_gc0g 17546   /_s cqus 17613  toInccipo 18637  Grpcgrp 19080  inv_gcminusg 19081  SubGrpcsubg 19266  NrmSGrpcnsg 19267   ~_QG cqg 19268  LSSumclsm 19784

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7372  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7865  df-1st 7988  df-2nd 7989  df-tpos 8226  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-rdg 8401  df-1o 8459  df-er 8700  df-ec 8702  df-qs 8706  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-inf 9417  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11489  df-neg 11490  df-nn 12280  df-2 12349  df-3 12350  df-4 12351  df-5 12352  df-6 12353  df-7 12354  df-8 12355  df-9 12356  df-n0 12551  df-z 12638  df-dec 12759  df-uz 12910  df-fz 13584  df-struct 17261  df-sets 17278  df-slot 17296  df-ndx 17308  df-base 17324  df-ress 17345  df-plusg 17377  df-mulr 17378  df-sca 17380  df-vsca 17381  df-ip 17382  df-tset 17383  df-ple 17384  df-ds 17386  df-0g 17548  df-imas 17616  df-qus 17617  df-mgm 18752  df-sgrp 18844  df-mnd 18860  df-grp 19083  df-minusg 19084  df-subg 19269  df-nsg 19270  df-eqg 19271  df-oppg 19496  df-lsm 19786

This theorem is used by:  nsgqusf1olem2  33873  nsgqusf1olem3  33874