Theorem nsgqusf1olem2 34021

Description: Lemma for nsgqusf1o 34023. (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
nsgqusf1olem2	⊢ (𝜑 → ran 𝐸 = 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 nsgqusf1olem2

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
2		nsgqusf1o.s	. . . . . . . . . 10 ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}
3	2	reqabi 3438	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑆 ↔ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
4		nsgqusf1o.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
5		nsgqusf1o.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (SubGrp‘𝑄)
6		nsgqusf1o.1	. . . . . . . . . . . 12 ⊢ ≤ = (le‘(toInc‘𝑆))
7		nsgqusf1o.2	. . . . . . . . . . . 12 ⊢ ≲ = (le‘(toInc‘𝑇))
8		nsgqusf1o.q	. . . . . . . . . . . 12 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
9		nsgqusf1o.p	. . . . . . . . . . . 12 ⊢ ⊕ = (LSSum‘𝐺)
10		nsgqusf1o.e	. . . . . . . . . . . 12 ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
11		nsgqusf1o.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
12		nsgqusf1o.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
13	4, 2, 5, 6, 7, 8, 9, 10, 11, 12	nsgqusf1olem1 34020	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
14	13	anasss 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15	14, 5	eleqtrdi 2874	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
16	3, 15	sylan2b 606	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
17	16	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
18	1, 17	eqeltrd 2864	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
19	18	r19.29an 3170	. . . . 5 ⊢ ((𝜑 ∧ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20		sseq2 3939	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑁 ⊆ ℎ ↔ 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
21	12	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
23	4, 8, 9, 21, 22	nsgmgclem 34018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
24	5	eleq2i 2856	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
25		nsgsubg 19383	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
26	12, 25	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
27	4	subgss 19352	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
28	26, 27	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
29	28	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
30	26	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
31	9	grplsmid 34011	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
32	30, 31	sylancom 600	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
33	24	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄))
34	8	nsgqus0 34017	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
35	12, 33, 34	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
36	35	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
37	32, 36	eqeltrd 2864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
38	29, 37	ssrabdv 4001	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
39	24, 38	sylan2br 607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
40	20, 23, 39	elrabd 3650	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ})
41	40, 2	eleqtrrdi 2875	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ 𝑆)
42		mpteq1 5197	. . . . . . . . 9 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
43	42	rneqd 5925	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
44	43	eqeq2d 2774	. . . . . . 7 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))))
45	44	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))))
46		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
47	46	subgss 19352	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
48	47	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
49	48	sselda 3913	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (Base‘𝑄))
50	8	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
51	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐵 = (Base‘𝐺))
52		ovexd 7465	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53		subgrcl 19356	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54	26, 53	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Grp)
55	54	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐺 ∈ Grp)
56	50, 51, 52, 55	qusbas 17732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
57	49, 56	eleqtrrd 2867	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58		elqsi 8800	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)) → ∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
59	57, 58	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
60		sneq 4594	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
61	60	oveq1d 7445	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
62	61	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
63		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝐵)
64		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
65	26	ad4antr 745	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
66	4, 9, 65, 63	quslsm 34012	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
67	64, 66	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁))
68		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 ∈ 𝑓)
69	67, 68	eqeltrrd 2865	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
70	62, 63, 69	elrabd 3650	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
71	70, 67	jca 521	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} ⊕ 𝑁)))
72	71	expl 463	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ((𝑥 ∈ 𝐵 ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} ⊕ 𝑁))))
73	72	reximdv2 3176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁) → ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)))
74	59, 73	mpd 16	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))
75		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
76	62	elrab 3648	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↔ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
77	75, 76	sylib 221	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
78		simpllr 788	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} ⊕ 𝑁))
79		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
80	78, 79	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 ∈ 𝑓)
81	80	anasss 472	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓)
82	81	adantllr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓)
83	77, 82	mpdan 700	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
84	83	r19.29an 3170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
85	74, 84	impbida 813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)))
86		eqid 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))
87	86	elrnmpt 5945	. . . . . . . . 9 ⊢ (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)))
88	87	elv 3459	. . . . . . . 8 ⊢ (𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))
89	85, 88	bitr4di 292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ 𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))))
90	89	eqrdv 2761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
91	41, 45, 90	rspcedvd 3582	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
92	19, 91	impbida 813	. . . 4 ⊢ (𝜑 → (∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
93	92	abbidv 2829	. . 3 ⊢ (𝜑 → {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)})
94	10	rnmpt 5944	. . 3 ⊢ ran 𝐸 = {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))}
95		abid1 2900	. . 3 ⊢ (SubGrp‘𝑄) = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)}
96	93, 94, 95	3eqtr4g 2823	. 2 ⊢ (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
97	96, 5	eqtr4di 2816	1 ⊢ (𝜑 → ran 𝐸 = 𝑇)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  {csn 4588   ↦ cmpt 5189  ran crn 5656  ‘cfv 6547  (class class class)co 7430  [cec 8729   / cqs 8730  Basecbs 17402  lecple 17450   /_s cqus 17692  toInccipo 18716  Grpcgrp 19159  SubGrpcsubg 19345  NrmSGrpcnsg 19346   ~_QG cqg 19347  LSSumclsm 19863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-tpos 8257  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-ec 8733  df-qs 8737  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-sup 9447  df-inf 9448  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-z 12707  df-dec 12828  df-uz 12979  df-fz 13653  df-struct 17340  df-sets 17357  df-slot 17375  df-ndx 17387  df-base 17403  df-ress 17424  df-plusg 17456  df-mulr 17457  df-sca 17459  df-vsca 17460  df-ip 17461  df-tset 17462  df-ple 17463  df-ds 17465  df-0g 17627  df-imas 17695  df-qus 17696  df-mgm 18831  df-sgrp 18923  df-mnd 18939  df-submnd 18994  df-grp 19162  df-minusg 19163  df-subg 19348  df-nsg 19349  df-eqg 19350  df-oppg 19575  df-lsm 19865

This theorem is used by:  nsgqusf1o  34023