Theorem nsgqusf1olem3 33724

Description: Lemma for nsgqusf1o 33725. (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 5948	. . . 4 ⊢ (ℎ ∈ V → (ℎ ∈ ran 𝐹 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
3	2	elv 3460	. . 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 3439	. . . 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 33722	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
15		eleq2 2852	. . . . . . . . . 10 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))))
16	15	rabbidv 3423	. . . . . . . . 9 ⊢ (𝑓 = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))})
17	16	eqeq2d 2774	. . . . . . . 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 486	. . . . . . 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 1944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵)
20		nfmpt1 5210	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
21	20	nfrn 5942	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
22	21	nfel2 2943	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
23	19, 22	nfan 1929	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
24		nsgsubg 19219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
25	13, 24	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
26		subgrcl 19192	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
27	25, 26	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ Grp)
28	27	ad4antr 744	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝐺 ∈ Grp)
29	28	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝐺 ∈ Grp)
30	6	subgss 19188	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵)
31	30	ad3antlr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) → ℎ ⊆ 𝐵)
32	31	sselda 3937	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ 𝐵)
33	32	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ 𝐵)
34		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑎 ∈ 𝐵)
35	34	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎 ∈ 𝐵)
36		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝐺) = (+g‘𝐺)
37		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (invg‘𝐺) = (invg‘𝐺)
38	6, 36, 37	grpasscan1 19063	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑥(+g‘𝐺)(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎)) = 𝑎)
39	29, 33, 35, 38	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (𝑥(+g‘𝐺)(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎)) = 𝑎)
40		simp-5r 797	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → ℎ ∈ (SubGrp‘𝐺))
41		simplr 780	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑥 ∈ ℎ)
42		simp-4r 795	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑁 ⊆ ℎ)
43	6	subgss 19188	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
44	25, 43	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
45	44	ad5antr 746	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑁 ⊆ 𝐵)
46		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁)
47	6, 46	eqger 19241	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑁) Er 𝐵)
48	25, 47	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ~QG 𝑁) Er 𝐵)
49	48	ad4antr 744	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → (𝐺 ~QG 𝑁) Er 𝐵)
50	49	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (𝐺 ~QG 𝑁) Er 𝐵)
51	49, 34	erth 8745	. . . . . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → 𝑁 ∈ (SubGrp‘𝐺))
53	6, 11, 52, 34	quslsm 33714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → [𝑎](𝐺 ~QG 𝑁) = ({𝑎} ⊕ 𝑁))
54	6, 11, 52, 32	quslsm 33714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁))
55	53, 54	eqeq12d 2779	. . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) → (𝑎(𝐺 ~QG 𝑁)𝑥 ↔ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)))
57	56	biimpar 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎(𝐺 ~QG 𝑁)𝑥)
58	50, 57	ersym 8703	. . . . . . . . . . . . . . . 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 19240	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) → (𝑥(𝐺 ~QG 𝑁)𝑎 ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎) ∈ 𝑁)))
60	59	biimpa 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) ∧ 𝑥(𝐺 ~QG 𝑁)𝑎) → (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎) ∈ 𝑁))
61	60	simp3d 1162	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ⊆ 𝐵) ∧ 𝑥(𝐺 ~QG 𝑁)𝑎) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎) ∈ 𝑁)
62	29, 45, 58, 61	syl21anc 850	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎) ∈ 𝑁)
63	42, 62	sseldd 3938	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎) ∈ ℎ)
64	36	subgcl 19197	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ ℎ ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎) ∈ ℎ) → (𝑥(+g‘𝐺)(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎)) ∈ ℎ)
65	40, 41, 63, 64	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → (𝑥(+g‘𝐺)(((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑎)) ∈ ℎ)
66	39, 65	eqeltrrd 2864	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑥 ∈ ℎ) ∧ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁)) → 𝑎 ∈ ℎ)
67	66	adantllr 731	. . . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
69		ovex 7443	. . . . . . . . . . . . 13 ⊢ ({𝑥} ⊕ 𝑁) ∈ V
70	68, 69	elrnmpti 5952	. . . . . . . . . . . 12 ⊢ (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
71	70	bilani 509	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
72	23, 67, 71	r19.29af 3274	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) → 𝑎 ∈ ℎ)
73		simpr 489	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → 𝑎 ∈ ℎ)
74		ovexd 7445	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ V)
75		sneq 4599	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
76	75	oveq1d 7425	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ({𝑥} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁))
77	76	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
78	77	adantl 486	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) ∧ 𝑥 = 𝑎) → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
79	68, 73, 74, 78	elrnmptdv 5955	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
80	72, 79	impbida 812	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) ∧ 𝑎 ∈ 𝐵) → (({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ↔ 𝑎 ∈ ℎ))
81	80	rabbidva 3422	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))} = {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ})
82	30	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) → ℎ ⊆ 𝐵)
83		dfss7 4204	. . . . . . . . . 10 ⊢ (ℎ ⊆ 𝐵 ↔ {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
84	82, 83	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) → {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
85	84	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → {𝑎 ∈ 𝐵 ∣ 𝑎 ∈ ℎ} = ℎ)
86	81, 85	eqtr2d 2799	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))})
87	14, 18, 86	rspcedvd 3583	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
88	87	anasss 471	. . . . 5 ⊢ ((𝜑 ∧ (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ)) → ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
89	13	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ (NrmSGrp‘𝐺))
90	7	eleq2i 2855	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrp‘𝑄))
91	90	bilani 509	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑓 ∈ (SubGrp‘𝑄))
92	6, 10, 11, 89, 91	nsgmgclem 33720	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
93	92	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
94		eleq1 2851	. . . . . . . . 9 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (ℎ ∈ (SubGrp‘𝐺) ↔ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺)))
95	94	adantl 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ↔ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺)))
96	93, 95	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → ℎ ∈ (SubGrp‘𝐺))
97	44	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
98	25	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
99		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
100	11	grplsmid 33713	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
101	98, 99, 100	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
102	10	nsgqus0 33719	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
103	89, 91, 102	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ 𝑓)
104	103	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
105	101, 104	eqeltrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
106	97, 105	ssrabdv 4027	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
107	106	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
108		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
109	107, 108	sseqtrrd 3974	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → 𝑁 ⊆ ℎ)
110	96, 109	jca 520	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
111	110	r19.29an 3169	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) → (ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ))
112	88, 111	impbida 812	. . . 4 ⊢ (𝜑 → ((ℎ ∈ (SubGrp‘𝐺) ∧ 𝑁 ⊆ ℎ) ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
113	5, 112	bitrid 286	. . 3 ⊢ (𝜑 → (ℎ ∈ 𝑆 ↔ ∃𝑓 ∈ 𝑇 ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
114	3, 113	bitr4id 293	. 2 ⊢ (𝜑 → (ℎ ∈ ran 𝐹 ↔ ℎ ∈ 𝑆))
115	114	eqrdv 2761	1 ⊢ (𝜑 → ran 𝐹 = 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  {csn 4589   class class class wbr 5109   ↦ cmpt 5192  ran crn 5662  ‘cfv 6536  (class class class)co 7410   Er wer 8687  [cec 8688  Basecbs 17264  +_gcplusg 17305  lecple 17312   /_s cqus 17554  toInccipo 18578  Grpcgrp 18995  inv_gcminusg 18996  SubGrpcsubg 19181  NrmSGrpcnsg 19182   ~_QG cqg 19183  LSSumclsm 19699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-ec 8692  df-qs 8696  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-0g 17489  df-imas 17557  df-qus 17558  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-submnd 18837  df-grp 18998  df-minusg 18999  df-subg 19184  df-nsg 19185  df-eqg 19186  df-oppg 19411  df-lsm 19701

This theorem is referenced by:  nsgqusf1o  33725