Theorem nsgqusf1olem2 32525

Description: Lemma for nsgqusf1o 32527. (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 486	. . . . . . 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 3455	. . . . . . . . 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 32524	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
14	13	anasss 468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15	14, 5	eleqtrdi 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
16	3, 15	sylan2b 595	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
17	16	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ (SubGrp‘𝑄))
18	1, 17	eqeltrd 2834	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
19	18	r19.29an 3159	. . . . 5 ⊢ ((𝜑 ∧ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20		sseq2 4009	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑁 ⊆ ℎ ↔ 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
21	12	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
23	4, 8, 9, 21, 22	nsgmgclem 32522	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
24	5	eleq2i 2826	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
25		nsgsubg 19038	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
26	12, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
27	4	subgss 19007	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
29	28	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
30	26	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
31	9	grplsmid 32514	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
32	30, 31	sylancom 589	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
33	24	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄))
34	8	nsgqus0 32521	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
35	12, 33, 34	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
36	35	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
37	32, 36	eqeltrd 2834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
38	29, 37	ssrabdv 4072	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
39	24, 38	sylan2br 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
40	20, 23, 39	elrabd 3686	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ})
41	40, 2	eleqtrrdi 2845	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ 𝑆)
42		mpteq1 5242	. . . . . . . . 9 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
43	42	rneqd 5938	. . . . . . . 8 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
44	43	eqeq2d 2744	. . . . . . 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 483	. . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
47	46	subgss 19007	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
48	47	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
49	48	sselda 3983	. . . . . . . . . . . 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 7444	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53		subgrcl 19011	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
54	26, 53	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Grp)
55	54	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝐺 ∈ Grp)
56	50, 51, 52, 55	qusbas 17491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
57	49, 56	eleqtrrd 2837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58		elqsi 8764	. . . . . . . . . . 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 4639	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
61	60	oveq1d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
62	61	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
63		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝐵)
64		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
65	26	ad4antr 731	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
66	4, 9, 65, 63	quslsm 32516	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
67	64, 66	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} ⊕ 𝑁))
68		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 ∈ 𝑓)
69	67, 68	eqeltrrd 2835	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
70	62, 63, 69	elrabd 3686	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 ∈ 𝑓) ∧ 𝑥 ∈ 𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
71	70, 67	jca 513	. . . . . . . . . . . 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 459	. . . . . . . . . . 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 3165	. . . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
76	62	elrab 3684	. . . . . . . . . . . 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 775	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} ⊕ 𝑁))
79		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
80	78, 79	eqeltrd 2834	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ 𝑥 ∈ 𝐵) ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓) → 𝑖 ∈ 𝑓)
81	80	anasss 468	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) ∧ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓)) → 𝑖 ∈ 𝑓)
82	81	adantllr 718	. . . . . . . . . . 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 686	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
84	83	r19.29an 3159	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)) → 𝑖 ∈ 𝑓)
85	74, 84	impbida 800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁)))
86		eqid 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))
87	86	elrnmpt 5956	. . . . . . . . 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 3481	. . . . . . . 8 ⊢ (𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}𝑖 = ({𝑥} ⊕ 𝑁))
89	85, 88	bitr4di 289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → (𝑖 ∈ 𝑓 ↔ 𝑖 ∈ ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁))))
90	89	eqrdv 2731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↦ ({𝑥} ⊕ 𝑁)))
91	41, 45, 90	rspcedvd 3615	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (SubGrp‘𝑄)) → ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
92	19, 91	impbida 800	. . . 4 ⊢ (𝜑 → (∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
93	92	abbidv 2802	. . 3 ⊢ (𝜑 → {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)})
94	10	rnmpt 5955	. . 3 ⊢ ran 𝐸 = {𝑓 ∣ ∃ℎ ∈ 𝑆 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))}
95		abid1 2871	. . 3 ⊢ (SubGrp‘𝑄) = {𝑓 ∣ 𝑓 ∈ (SubGrp‘𝑄)}
96	93, 94, 95	3eqtr4g 2798	. 2 ⊢ (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
97	96, 5	eqtr4di 2791	1 ⊢ (𝜑 → ran 𝐸 = 𝑇)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  {crab 3433  Vcvv 3475   ⊆ wss 3949  {csn 4629   ↦ cmpt 5232  ran crn 5678  ‘cfv 6544  (class class class)co 7409  [cec 8701   / cqs 8702  Basecbs 17144  lecple 17204   /_s cqus 17451  toInccipo 18480  Grpcgrp 18819  SubGrpcsubg 19000  NrmSGrpcnsg 19001   ~_QG cqg 19002  LSSumclsm 19502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-ec 8705  df-qs 8709  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-0g 17387  df-imas 17454  df-qus 17455  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-subg 19003  df-nsg 19004  df-eqg 19005  df-oppg 19210  df-lsm 19504

This theorem is referenced by:  nsgqusf1o  32527