Theorem nsgqusf1olem2 33667

Description: Lemma for nsgqusf1o 33669. (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 489	. . . . . . 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 3446	. . . . . . . . 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 33666	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
14	13	anasss 471	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15	14, 5	eleqtrdi 2879	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
16	3, 15	sylan2b 605	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
17	16	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
18	1, 17	eqeltrd 2869	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
19	18	r19.29an 3175	. . . . 5 ⊢ ((𝜑 ∧ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20		sseq2 3971	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑁 ⊆ ℎ ↔ 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
21	12	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
23	4, 8, 9, 21, 22	nsgmgclem 33664	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
24	5	eleq2i 2861	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
25		nsgsubg 19224	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
26	12, 25	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
27	4	subgss 19193	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
28	26, 27	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
29	28	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
30	26	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
31	9	grplsmid 33657	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
32	30, 31	sylancom 599	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
33	24	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄))
34	8	nsgqus0 33663	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
35	12, 33, 34	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
36	35	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
37	32, 36	eqeltrd 2869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
38	29, 37	ssrabdv 4035	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
39	24, 38	sylan2br 606	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
40	20, 23, 39	elrabd 3661	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ})
41	40, 2	eleqtrrdi 2880	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ 𝑆)
42		mpteq1 5204	. . . . . . . . 9 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
43	42	rneqd 5929	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
44	43	eqeq2d 2780	. . . . . . 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 486	. . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
47	46	subgss 19193	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
48	47	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
49	48	sselda 3945	. . . . . . . . . . . 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 7446	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53		subgrcl 19197	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54	26, 53	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Grp)
55	54	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐺 ∈ Grp)
56	50, 51, 52, 55	qusbas 17599	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
57	49, 56	eleqtrrd 2872	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58		elqsi 8763	. . . . . . . . . . 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 4604	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
61	60	oveq1d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
62	61	eleq1d 2854	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
63		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝐵)
64		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
65	26	ad4antr 744	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
66	4, 9, 65, 63	quslsm 33658	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
67	64, 66	eqtrd 2804	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁))
68		simpllr 787	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 ∈ 𝑓)
69	67, 68	eqeltrrd 2870	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
70	62, 63, 69	elrabd 3661	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
71	70, 67	jca 520	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 3181	. . . . . . . . . 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 780	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
76	62	elrab 3659	. . . . . . . . . . . 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 787	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} ⊕ 𝑁))
79		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
80	78, 79	eqeltrd 2869	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 ∈ 𝑓)
81	80	anasss 471	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓)
82	81	adantllr 731	. . . . . . . . . . 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 699	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
84	83	r19.29an 3175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
85	74, 84	impbida 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)))
86		eqid 2769	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))
87	86	elrnmpt 5949	. . . . . . . . 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 3468	. . . . . . . 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 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
91	41, 45, 90	rspcedvd 3592	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
92	19, 91	impbida 812	. . . 4 ⊢ (𝜑 → (∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
93	92	abbidv 2835	. . 3 ⊢ (𝜑 → {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)})
94	10	rnmpt 5948	. . 3 ⊢ ran 𝐸 = {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))}
95		abid1 2905	. . 3 ⊢ (SubGrp‘𝑄) = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)}
96	93, 94, 95	3eqtr4g 2829	. 2 ⊢ (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
97	96, 5	eqtr4di 2822	1 ⊢ (𝜑 → ran 𝐸 = 𝑇)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  {csn 4594   ↦ cmpt 5196  ran crn 5663  ‘cfv 6537  (class class class)co 7411  [cec 8692   / cqs 8693  Basecbs 17269  lecple 17317   /_s cqus 17559  toInccipo 18583  Grpcgrp 19000  SubGrpcsubg 19186  NrmSGrpcnsg 19187   ~_QG cqg 19188  LSSumclsm 19704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-ec 8696  df-qs 8700  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-inf 9403  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-0g 17494  df-imas 17562  df-qus 17563  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-submnd 18842  df-grp 19003  df-minusg 19004  df-subg 19189  df-nsg 19190  df-eqg 19191  df-oppg 19416  df-lsm 19706

This theorem is referenced by:  nsgqusf1o  33669