nsgqusf1olem3 - Mathbox for Thierry Arnoux

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Theorem nsgqusf1olem3 31502

Description: Lemma for nsgqusf1o 31503. (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
nsgqusf1olem3	⊢ (𝜑 → ran 𝐹 = 𝑆)

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 nsgqusf1olem3**
Step	Hyp	Ref	Expression
1		nsgqusf1o.f	. . . . 5 ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
2	1	elrnmpt 5854	. . . 4 ⊢ (ℎ ∈ V → (ℎ ∈ ran 𝐹 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
3	2	elv 3428	. . 3 ⊢ (ℎ ∈ ran 𝐹 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
4		nsgqusf1o.s	. . . . 5 ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}
5	4	rabeq2i 3412	. . . 4 ⊢ (ℎ ∈ 𝑆 ↔ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
6		nsgqusf1o.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
7		nsgqusf1o.t	. . . . . . . 8 ⊢ 𝑇 = (SubGrp‘𝑄)
8		nsgqusf1o.1	. . . . . . . 8 ⊢ ≤ = (le‘(toInc‘𝑆))
9		nsgqusf1o.2	. . . . . . . 8 ⊢ ≲ = (le‘(toInc‘𝑇))
10		nsgqusf1o.q	. . . . . . . 8 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁))
11		nsgqusf1o.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
12		nsgqusf1o.e	. . . . . . . 8 ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
13		nsgqusf1o.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
14	6, 4, 7, 8, 9, 10, 11, 12, 1, 13	nsgqusf1olem1 31500	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
16	15	rabbidv 3404	. . . . . . . . 9 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))})
17	16	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↔ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))}))
18	17	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↔ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))}))
19		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵)
20		nfmpt1 5178	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
21	20	nfrn 5850	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
22	21	nfel2 2924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
23	19, 22	nfan 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
24		nsgsubg 18701	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
25	13, 24	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
26		subgrcl 18675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ Grp)
28	27	ad4antr 728	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝐺 ∈ Grp)
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝐺 ∈ Grp)
30	6	subgss 18671	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵)
31	30	ad3antlr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) → ℎ ⊆ 𝐵)
32	31	sselda 3917	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ 𝐵)
33	32	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ 𝐵)
34		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑎 ∈ 𝐵)
35	34	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎 ∈ 𝐵)
36		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
37		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
38	6, 36, 37	grpasscan1 18553	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑥(+_g‘𝐺)(((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎)) = 𝑎)
39	29, 33, 35, 38	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (𝑥(+_g‘𝐺)(((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎)) = 𝑎)
40		simp-5r 782	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
41		simplr 765	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ ℎ)
42		simp-4r 780	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑁 ⊆ ℎ)
43	6	subgss 18671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
44	25, 43	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
45	44	ad5antr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑁 ⊆ 𝐵)
46		eqid 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
47	6, 46	eqger 18721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er 𝐵)
48	25, 47	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er 𝐵)
49	48	ad4antr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → (𝐺 ~_QG 𝑁) Er 𝐵)
50	49	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (𝐺 ~_QG 𝑁) Er 𝐵)
51	49, 34	erth 8505	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → (𝑎(𝐺 ~_QG 𝑁)𝑥 ↔ [𝑎](𝐺 ~_QG 𝑁) = [𝑥](𝐺 ~_QG 𝑁)))
52	25	ad4antr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑁 ∈ (SubGrp‘𝐺))
53	6, 11, 52, 34	quslsm 31495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → [𝑎](𝐺 ~_QG 𝑁) = ({𝑎} ⊕ 𝑁))
54	6, 11, 52, 32	quslsm 31495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~_QG 𝑁) = ({𝑥} ⊕ 𝑁))
55	53, 54	eqeq12d 2754	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → ([𝑎](𝐺 ~_QG 𝑁) = [𝑥](𝐺 ~_QG 𝑁) ↔ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)))
56	51, 55	bitrd 278	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → (𝑎(𝐺 ~_QG 𝑁)𝑥 ↔ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)))
57	56	biimpar 477	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎(𝐺 ~_QG 𝑁)𝑥)
58	50, 57	ersym 8468	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑥(𝐺 ~_QG 𝑁)𝑎)
59	6, 37, 36, 46	eqgval 18720	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) → (𝑥(𝐺 ~_QG 𝑁)𝑎 ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ 𝑁)))
60	59	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) ∧ 𝑥(𝐺 ~_QG 𝑁)𝑎) → (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ 𝑁))
61	60	simp3d 1142	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) ∧ 𝑥(𝐺 ~_QG 𝑁)𝑎) → (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ 𝑁)
62	29, 45, 58, 61	syl21anc 834	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ 𝑁)
63	42, 62	sseldd 3918	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ ℎ)
64	36	subgcl 18680	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ ∧ (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ ℎ) → (𝑥(+_g‘𝐺)(((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎)) ∈ ℎ)
65	40, 41, 63, 64	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (𝑥(+_g‘𝐺)(((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎)) ∈ ℎ)
66	39, 65	eqeltrrd 2840	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎 ∈ ℎ)
67	66	adantllr 715	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎 ∈ ℎ)
68		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
69		ovex 7288	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊕ 𝑁) ∈ V
70	68, 69	elrnmpti 5858	. . . . . . . . . . . . 13 ⊢ (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
71	70	biimpi 215	. . . . . . . . . . . 12 ⊢ (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
72	71	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
73	23, 67, 72	r19.29af 3259	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑎 ∈ ℎ)
74		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → 𝑎 ∈ ℎ)
75		ovexd 7290	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ V)
76		sneq 4568	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
77	76	oveq1d 7270	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ({𝑥} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁))
78	77	eqcomd 2744	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
79	78	adantl 481	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) ∧ 𝑥 = 𝑎) → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
80	68, 74, 75, 79	elrnmptdv 5860	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
81	73, 80	impbida 797	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) → (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑎 ∈ ℎ))
82	81	rabbidva 3402	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ})
83	30	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) → ℎ ⊆ 𝐵)
84		dfss7 4171	. . . . . . . . . 10 ⊢ (ℎ ⊆ 𝐵 ↔ {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
85	83, 84	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) → {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
86	85	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
87	82, 86	eqtr2d 2779	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))})
88	14, 18, 87	rspcedvd 3555	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
89	88	anasss 466	. . . . 5 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
90	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ (NrmSGrp‘𝐺))
91	7	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
92	91	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄))
93	92	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑓 ∈ (SubGrp‘𝑄))
94	6, 10, 11, 90, 93	nsgmgclem 31498	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
95	94	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
96		eleq1 2826	. . . . . . . . 9 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (ℎ ∈ (SubGrp‘𝐺) ↔ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺)))
97	96	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ↔ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺)))
98	95, 97	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → ℎ ∈ (SubGrp‘𝐺))
99	44	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
100	25	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
101		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
102	11	grplsmid 31494	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
103	100, 101, 102	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
104	10	nsgqus0 31497	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
105	90, 93, 104	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
106	105	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
107	103, 106	eqeltrd 2839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
108	99, 107	ssrabdv 4003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
109	108	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
110		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
111	109, 110	sseqtrrd 3958	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → 𝑁 ⊆ ℎ)
112	98, 111	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
113	112	r19.29an 3216	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
114	89, 113	impbida 797	. . . 4 ⊢ (𝜑 → ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ) ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
115	5, 114	syl5bb 282	. . 3 ⊢ (𝜑 → (ℎ ∈ 𝑆 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
116	3, 115	bitr4id 289	. 2 ⊢ (𝜑 → (ℎ ∈ ran 𝐹 ↔ ℎ ∈ 𝑆))
117	116	eqrdv 2736	1 ⊢ (𝜑 → ran 𝐹 = 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 {crab 3067 Vcvv 3422 ⊆ wss 3883 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ran crn 5581 ‘cfv 6418 (class class class)co 7255 Er wer 8453 [cec 8454 Basecbs 16840 +_gcplusg 16888 lecple 16895 /_s cqus 17133 toInccipo 18160 Grpcgrp 18492 inv_gcminusg 18493 SubGrpcsubg 18664 NrmSGrpcnsg 18665 ~_QG cqg 18666 LSSumclsm 19154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-ec 8458 df-qs 8462 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-0g 17069 df-imas 17136 df-qus 17137 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-subg 18667 df-nsg 18668 df-eqg 18669 df-oppg 18865 df-lsm 19156

This theorem is referenced by: nsgqusf1o 31503

W3C validator