Theorem nsgqusf1olem2 32192

Description: Lemma for nsgqusf1o 32194. (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 485	. . . . . . 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 3429	. . . . . . . . 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 32191	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
14	13	anasss 467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15	14, 5	eleqtrdi 2848	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
16	3, 15	sylan2b 594	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
17	16	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
18	1, 17	eqeltrd 2838	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
19	18	r19.29an 3155	. . . . 5 ⊢ ((𝜑 ∧ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20		sseq2 3970	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑁 ⊆ ℎ ↔ 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
21	12	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
23	4, 8, 9, 21, 22	nsgmgclem 32189	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
24	5	eleq2i 2829	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
25		nsgsubg 18960	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
26	12, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
27	4	subgss 18929	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
29	28	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
30	26	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
31	9	grplsmid 32185	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
32	30, 31	sylancom 588	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
33	24	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄))
34	8	nsgqus0 32188	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
35	12, 33, 34	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
36	35	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
37	32, 36	eqeltrd 2838	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
38	29, 37	ssrabdv 4031	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
39	24, 38	sylan2br 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
40	20, 23, 39	elrabd 3647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ})
41	40, 2	eleqtrrdi 2849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ 𝑆)
42		mpteq1 5198	. . . . . . . . 9 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
43	42	rneqd 5893	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
44	43	eqeq2d 2747	. . . . . . 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 482	. . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
47	46	subgss 18929	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
48	47	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
49	48	sselda 3944	. . . . . . . . . . . 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 7392	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53		subgrcl 18933	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54	26, 53	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Grp)
55	54	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐺 ∈ Grp)
56	50, 51, 52, 55	qusbas 17427	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
57	49, 56	eleqtrrd 2841	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58		elqsi 8709	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)) → ∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ∃𝑥 ∈ 𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
60		sneq 4596	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
61	60	oveq1d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
62	61	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
63		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝐵)
64		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
65	26	ad4antr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
66	4, 9, 65, 63	quslsm 32186	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
67	64, 66	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁))
68		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 ∈ 𝑓)
69	67, 68	eqeltrrd 2839	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
70	62, 63, 69	elrabd 3647	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
71	70, 67	jca 512	. . . . . . . . . . . 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 458	. . . . . . . . . . 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 3161	. . . . . . . . . 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 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))
75		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
76	62	elrab 3645	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↔ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
77	75, 76	sylib 217	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
78		simpllr 774	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} ⊕ 𝑁))
79		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
80	78, 79	eqeltrd 2838	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 ∈ 𝑓)
81	80	anasss 467	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓)
82	81	adantllr 717	. . . . . . . . . . 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 685	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
84	83	r19.29an 3155	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
85	74, 84	impbida 799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)))
86		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))
87	86	elrnmpt 5911	. . . . . . . . 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 3451	. . . . . . . 8 ⊢ (𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))
89	85, 88	bitr4di 288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ 𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))))
90	89	eqrdv 2734	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
91	41, 45, 90	rspcedvd 3583	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
92	19, 91	impbida 799	. . . 4 ⊢ (𝜑 → (∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
93	92	abbidv 2805	. . 3 ⊢ (𝜑 → {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)})
94	10	rnmpt 5910	. . 3 ⊢ ran 𝐸 = {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))}
95		abid1 2874	. . 3 ⊢ (SubGrp‘𝑄) = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)}
96	93, 94, 95	3eqtr4g 2801	. 2 ⊢ (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
97	96, 5	eqtr4di 2794	1 ⊢ (𝜑 → ran 𝐸 = 𝑇)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910  {csn 4586   ↦ cmpt 5188  ran crn 5634  ‘cfv 6496  (class class class)co 7357  [cec 8646   / cqs 8647  Basecbs 17083  lecple 17140   /_s cqus 17387  toInccipo 18416  Grpcgrp 18748  SubGrpcsubg 18922  NrmSGrpcnsg 18923   ~_QG cqg 18924  LSSumclsm 19416

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-0g 17323  df-imas 17390  df-qus 17391  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-subg 18925  df-nsg 18926  df-eqg 18927  df-oppg 19124  df-lsm 19418

This theorem is referenced by:  nsgqusf1o  32194