nsgqusf1olem3 - Mathbox for Thierry Arnoux

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Theorem nsgqusf1olem3 33475

Description: Lemma for nsgqusf1o 33476. (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 5913	. . . 4 ⊢ (ℎ ∈ V → (ℎ ∈ ran 𝐹 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
3	2	elv 3434	. . 3 ⊢ (ℎ ∈ ran 𝐹 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
4		nsgqusf1o.s	. . . . 5 ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}
5	4	reqabi 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 33473	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15		eleq2 2825	. . . . . . . . . 10 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
16	15	rabbidv 3396	. . . . . . . . 9 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))})
17	16	eqeq2d 2747	. . . . . . . 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 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵)
20		nfmpt1 5184	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
21	20	nfrn 5907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
22	21	nfel2 2917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
23	19, 22	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
24		nsgsubg 19133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
25	13, 24	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
26		subgrcl 19107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ Grp)
28	27	ad4antr 733	. . . . . . . . . . . . . . 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 19103	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵)
31	30	ad3antlr 732	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) → ℎ ⊆ 𝐵)
32	31	sselda 3921	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
37		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
38	6, 36, 37	grpasscan1 18977	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑥(+_g‘𝐺)(((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎)) = 𝑎)
39	29, 33, 35, 38	syl3anc 1374	. . . . . . . . . . . . 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 786	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
41		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ ℎ)
42		simp-4r 784	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑁 ⊆ ℎ)
43	6	subgss 19103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
44	25, 43	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
45	44	ad5antr 735	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑁 ⊆ 𝐵)
46		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
47	6, 46	eqger 19153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er 𝐵)
48	25, 47	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er 𝐵)
49	48	ad4antr 733	. . . . . . . . . . . . . . . . . 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 8698	. . . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑁 ∈ (SubGrp‘𝐺))
53	6, 11, 52, 34	quslsm 33465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → [𝑎](𝐺 ~_QG 𝑁) = ({𝑎} ⊕ 𝑁))
54	6, 11, 52, 32	quslsm 33465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~_QG 𝑁) = ({𝑥} ⊕ 𝑁))
55	53, 54	eqeq12d 2752	. . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . . 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 8656	. . . . . . . . . . . . . . . 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 19152	. . . . . . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) ∧ 𝑥(𝐺 ~_QG 𝑁)𝑎) → (((inv_g‘𝐺)‘𝑥)(+_g‘𝐺)𝑎) ∈ 𝑁)
62	29, 45, 58, 61	syl21anc 838	. . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 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 19112	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 2837	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎 ∈ ℎ)
67	66	adantllr 720	. . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
69		ovex 7400	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊕ 𝑁) ∈ V
70	68, 69	elrnmpti 5917	. . . . . . . . . . . . 13 ⊢ (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
71	70	biimpi 216	. . . . . . . . . . . 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 3246	. . . . . . . . . 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 7402	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ V)
76		sneq 4577	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
77	76	oveq1d 7382	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ({𝑥} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁))
78	77	eqcomd 2742	. . . . . . . . . . . 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 5920	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
81	73, 80	impbida 801	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) → (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑎 ∈ ℎ))
82	81	rabbidva 3395	. . . . . . . 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 4191	. . . . . . . . . 10 ⊢ (ℎ ⊆ 𝐵 ↔ {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
85	83, 84	sylib 218	. . . . . . . . 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 2772	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))})
88	14, 18, 87	rspcedvd 3566	. . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
92	91	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑇 → 𝑓 ∈ (SubGrp‘𝑄))
93	92	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑓 ∈ (SubGrp‘𝑄))
94	6, 10, 11, 90, 93	nsgmgclem 33471	. . . . . . . . 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 2824	. . . . . . . . 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 257	. . . . . . 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 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
101		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
102	11	grplsmid 33464	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
103	100, 101, 102	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
104	10	nsgqus0 33470	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
105	90, 93, 104	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
106	105	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
107	103, 106	eqeltrd 2836	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
108	99, 107	ssrabdv 4013	. . . . . . . . 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 3959	. . . . . . 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 3141	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
114	89, 113	impbida 801	. . . 4 ⊢ (𝜑 → ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ) ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
115	5, 114	bitrid 283	. . 3 ⊢ (𝜑 → (ℎ ∈ 𝑆 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
116	3, 115	bitr4id 290	. 2 ⊢ (𝜑 → (ℎ ∈ ran 𝐹 ↔ ℎ ∈ 𝑆))
117	116	eqrdv 2734	1 ⊢ (𝜑 → ran 𝐹 = 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 {crab 3389 Vcvv 3429 ⊆ wss 3889 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 ran crn 5632 ‘cfv 6498 (class class class)co 7367 Er wer 8640 [cec 8641 Basecbs 17179 +_gcplusg 17220 lecple 17227 /_s cqus 17469 toInccipo 18493 Grpcgrp 18909 inv_gcminusg 18910 SubGrpcsubg 19096 NrmSGrpcnsg 19097 ~_QG cqg 19098 LSSumclsm 19609

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-ec 8645 df-qs 8649 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-0g 17404 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-subg 19099 df-nsg 19100 df-eqg 19101 df-oppg 19321 df-lsm 19611

This theorem is referenced by: nsgqusf1o 33476

W3C validator