Theorem nsgmgc 33349

Description: There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)

Hypotheses

Ref	Expression
nsgmgc.b	⊢ 𝐵 = (Base‘𝐺)
nsgmgc.s	⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}
nsgmgc.t	⊢ 𝑇 = (SubGrp‘𝑄)
nsgmgc.j	⊢ 𝐽 = (𝑉MGalConn𝑊)
nsgmgc.v	⊢ 𝑉 = (toInc‘𝑆)
nsgmgc.w	⊢ 𝑊 = (toInc‘𝑇)
nsgmgc.q	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
nsgmgc.p	⊢ ⊕ = (LSSum‘𝐺)
nsgmgc.e	⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
nsgmgc.f	⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
nsgmgc.n	⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))

Assertion

Ref	Expression
nsgmgc	⊢ (𝜑 → 𝐸𝐽𝐹)

Distinct variable groups:   ⊕ ,𝑎,ℎ,𝑥   𝐵,𝑎,ℎ,𝑥   𝐸,𝑎,𝑓,ℎ,𝑥   𝑓,𝐹,ℎ,𝑥   𝐺,𝑎,𝑓,ℎ,𝑥   𝑁,𝑎,ℎ,𝑥   𝑄,𝑎,𝑓,ℎ,𝑥   𝑆,𝑎,𝑓,ℎ,𝑥   𝑇,𝑎,𝑓,ℎ,𝑥   𝑓,𝑉,ℎ   𝑓,𝑊,ℎ   𝜑,𝑎,𝑓,ℎ,𝑥

Allowed substitution hints:   𝐵(𝑓)   ⊕ (𝑓)   𝐹(𝑎)   𝐽(𝑥,𝑓,ℎ,𝑎)   𝑁(𝑓)   𝑉(𝑥,𝑎)   𝑊(𝑥,𝑎)

Proof of Theorem nsgmgc

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1914	. . . . 5 ⊢ Ⅎℎ𝜑
2		vex 3440	. . . . . . . 8 ⊢ ℎ ∈ V
3	2	mptex 7159	. . . . . . 7 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V
4	3	rnex 7843	. . . . . 6 ⊢ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V)
6		nsgmgc.e	. . . . 5 ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
7	1, 5, 6	fnmptd 6623	. . . 4 ⊢ (𝜑 → 𝐸 Fn 𝑆)
8		nsgmgc.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
9		nsgmgc.q	. . . . . . . 8 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
10		nsgmgc.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
11		mpteq1 5181	. . . . . . . . . . 11 ⊢ (ℎ = 𝑘 → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ 𝑘 ↦ ({𝑥} ⊕ 𝑁)))
12	11	rneqd 5880	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝑘 ↦ ({𝑥} ⊕ 𝑁)))
13	12	cbvmptv 5196	. . . . . . . . 9 ⊢ (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) = (𝑘 ∈ 𝑆 ↦ ran (𝑥 ∈ 𝑘 ↦ ({𝑥} ⊕ 𝑁)))
14	6, 13	eqtri 2752	. . . . . . . 8 ⊢ 𝐸 = (𝑘 ∈ 𝑆 ↦ ran (𝑥 ∈ 𝑘 ↦ ({𝑥} ⊕ 𝑁)))
15		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))
16		nsgmgc.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → 𝑁 ∈ (NrmSGrp‘𝐺))
18		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ℎ ∈ 𝑆)
19		nsgmgc.s	. . . . . . . . . 10 ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}
20	19	ssrab3 4033	. . . . . . . . 9 ⊢ 𝑆 ⊆ (SubGrp‘𝐺)
21	20	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → 𝑆 ⊆ (SubGrp‘𝐺))
22	8, 9, 10, 14, 15, 17, 18, 21	qusima 33345	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → (𝐸‘ℎ) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) “ ℎ))
23	8, 9, 15	qusghm 19134	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ∈ (𝐺 GrpHom 𝑄))
24	17, 23	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ∈ (𝐺 GrpHom 𝑄))
25	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺))
26	25	sselda 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ℎ ∈ (SubGrp‘𝐺))
27		ghmima 19116	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ∈ (𝐺 GrpHom 𝑄) ∧ ℎ ∈ (SubGrp‘𝐺)) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) “ ℎ) ∈ (SubGrp‘𝑄))
28	24, 26, 27	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) “ ℎ) ∈ (SubGrp‘𝑄))
29	22, 28	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → (𝐸‘ℎ) ∈ (SubGrp‘𝑄))
30		nsgmgc.t	. . . . . 6 ⊢ 𝑇 = (SubGrp‘𝑄)
31	29, 30	eleqtrrdi 2839	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → (𝐸‘ℎ) ∈ 𝑇)
32	31	ralrimiva 3121	. . . 4 ⊢ (𝜑 → ∀ℎ ∈ 𝑆 (𝐸‘ℎ) ∈ 𝑇)
33		ffnfv 7053	. . . 4 ⊢ (𝐸:𝑆⟶𝑇 ↔ (𝐸 Fn 𝑆 ∧ ∀ℎ ∈ 𝑆 (𝐸‘ℎ) ∈ 𝑇))
34	7, 32, 33	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐸:𝑆⟶𝑇)
35		sseq2 3962	. . . . . 6 ⊢ (ℎ = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → (𝑁 ⊆ ℎ ↔ 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}))
36	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ∈ (NrmSGrp‘𝐺))
37		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑓 ∈ 𝑇)
38	37, 30	eleqtrdi 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑓 ∈ (SubGrp‘𝑄))
39	8, 9, 10, 36, 38	nsgmgclem 33348	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
40		nsgsubg 19037	. . . . . . . . . 10 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
41	16, 40	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
42	8	subgss 19006	. . . . . . . . 9 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝐵)
43	41, 42	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ⊆ 𝐵)
44	43	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ 𝐵)
45	41	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
46		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑎 ∈ 𝑁)
47	10	grplsmid 33341	. . . . . . . . 9 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
48	45, 46, 47	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) = 𝑁)
49	16	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ (NrmSGrp‘𝐺))
50	38	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑓 ∈ (SubGrp‘𝑄))
51	9	nsgqus0 33347	. . . . . . . . 9 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝑓)
52	49, 50, 51	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → 𝑁 ∈ 𝑓)
53	48, 52	eqeltrd 2828	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑇) ∧ 𝑎 ∈ 𝑁) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
54	44, 53	ssrabdv 4025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → 𝑁 ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
55	35, 39, 54	elrabd 3650	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ})
56	55, 19	eleqtrrdi 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ 𝑆)
57		nsgmgc.f	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
58	56, 57	fmptd 7048	. . 3 ⊢ (𝜑 → 𝐹:𝑇⟶𝑆)
59	34, 58	jca 511	. 2 ⊢ (𝜑 → (𝐸:𝑆⟶𝑇 ∧ 𝐹:𝑇⟶𝑆))
60	8	subgss 19006	. . . . . . . . . 10 ⊢ (ℎ ∈ (SubGrp‘𝐺) → ℎ ⊆ 𝐵)
61	26, 60	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ℎ ⊆ 𝐵)
62	61	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) → ℎ ⊆ 𝐵)
63	6	fvmpt2 6941	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ 𝑆 ∧ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V) → (𝐸‘ℎ) = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
64	18, 4, 63	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → (𝐸‘ℎ) = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
65	64	ad3antrrr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → (𝐸‘ℎ) = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
66		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → (𝐸‘ℎ) ⊆ 𝑓)
67	65, 66	eqsstrrd 3971	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ 𝑓)
68		eqid 2729	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))
69		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → 𝑎 ∈ ℎ)
70		sneq 4587	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎})
71	70	oveq1d 7364	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ({𝑥} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁))
72	71	eqeq2d 2740	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁) ↔ ({𝑎} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁)))
73	72	adantl 481	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) ∧ 𝑥 = 𝑎) → (({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁) ↔ ({𝑎} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁)))
74		eqidd 2730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) = ({𝑎} ⊕ 𝑁))
75	69, 73, 74	rspcedvd 3579	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → ∃𝑥 ∈ ℎ ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
76		ovexd 7384	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ V)
77	68, 75, 76	elrnmptd 5905	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
78	67, 77	sseldd 3936	. . . . . . . 8 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) ∧ 𝑎 ∈ ℎ) → ({𝑎} ⊕ 𝑁) ∈ 𝑓)
79	62, 78	ssrabdv 4025	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) → ℎ ⊆ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
80		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → 𝑓 ∈ 𝑇)
81	8	fvexi 6836	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
82	81	rabex 5278	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ V
83	57	fvmpt2 6941	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑇 ∧ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ∈ V) → (𝐹‘𝑓) = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
84	80, 82, 83	sylancl 586	. . . . . . . 8 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → (𝐹‘𝑓) = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
85	84	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) → (𝐹‘𝑓) = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
86	79, 85	sseqtrrd 3973	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ (𝐸‘ℎ) ⊆ 𝑓) → ℎ ⊆ (𝐹‘𝑓))
87	64	ad2antrr 726	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) → (𝐸‘ℎ) = ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))
88		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) → ℎ ⊆ (𝐹‘𝑓))
89	88	sselda 3935	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ (𝐹‘𝑓))
90	84	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) ∧ 𝑥 ∈ ℎ) → (𝐹‘𝑓) = {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
91	89, 90	eleqtrd 2830	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) ∧ 𝑥 ∈ ℎ) → 𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})
92		sneq 4587	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → {𝑎} = {𝑥})
93	92	oveq1d 7364	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → ({𝑎} ⊕ 𝑁) = ({𝑥} ⊕ 𝑁))
94	93	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (({𝑎} ⊕ 𝑁) ∈ 𝑓 ↔ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
95	94	elrab 3648	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} ↔ (𝑥 ∈ 𝐵 ∧ ({𝑥} ⊕ 𝑁) ∈ 𝑓))
96	95	simprbi 496	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓} → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
97	91, 96	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) ∧ 𝑥 ∈ ℎ) → ({𝑥} ⊕ 𝑁) ∈ 𝑓)
98	97	ralrimiva 3121	. . . . . . . 8 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) → ∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ 𝑓)
99	68	rnmptss 7057	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℎ ({𝑥} ⊕ 𝑁) ∈ 𝑓 → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ 𝑓)
100	98, 99	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ⊆ 𝑓)
101	87, 100	eqsstrd 3970	. . . . . 6 ⊢ ((((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) ∧ ℎ ⊆ (𝐹‘𝑓)) → (𝐸‘ℎ) ⊆ 𝑓)
102	86, 101	impbida 800	. . . . 5 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → ((𝐸‘ℎ) ⊆ 𝑓 ↔ ℎ ⊆ (𝐹‘𝑓)))
103	30	fvexi 6836	. . . . . 6 ⊢ 𝑇 ∈ V
104		nsgmgc.w	. . . . . . 7 ⊢ 𝑊 = (toInc‘𝑇)
105		eqid 2729	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
106	104, 105	ipole 18440	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ (𝐸‘ℎ) ∈ 𝑇 ∧ 𝑓 ∈ 𝑇) → ((𝐸‘ℎ)(le‘𝑊)𝑓 ↔ (𝐸‘ℎ) ⊆ 𝑓))
107	103, 31, 80, 106	mp3an2ani 1470	. . . . 5 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → ((𝐸‘ℎ)(le‘𝑊)𝑓 ↔ (𝐸‘ℎ) ⊆ 𝑓))
108		fvex 6835	. . . . . . 7 ⊢ (SubGrp‘𝐺) ∈ V
109	19, 108	rabex2 5280	. . . . . 6 ⊢ 𝑆 ∈ V
110	58	ffvelcdmda 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → (𝐹‘𝑓) ∈ 𝑆)
111	110	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → (𝐹‘𝑓) ∈ 𝑆)
112		nsgmgc.v	. . . . . . 7 ⊢ 𝑉 = (toInc‘𝑆)
113		eqid 2729	. . . . . . 7 ⊢ (le‘𝑉) = (le‘𝑉)
114	112, 113	ipole 18440	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ ℎ ∈ 𝑆 ∧ (𝐹‘𝑓) ∈ 𝑆) → (ℎ(le‘𝑉)(𝐹‘𝑓) ↔ ℎ ⊆ (𝐹‘𝑓)))
115	109, 18, 111, 114	mp3an2ani 1470	. . . . 5 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → (ℎ(le‘𝑉)(𝐹‘𝑓) ↔ ℎ ⊆ (𝐹‘𝑓)))
116	102, 107, 115	3bitr4d 311	. . . 4 ⊢ (((𝜑 ∧ ℎ ∈ 𝑆) ∧ 𝑓 ∈ 𝑇) → ((𝐸‘ℎ)(le‘𝑊)𝑓 ↔ ℎ(le‘𝑉)(𝐹‘𝑓)))
117	116	anasss 466	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ 𝑆 ∧ 𝑓 ∈ 𝑇)) → ((𝐸‘ℎ)(le‘𝑊)𝑓 ↔ ℎ(le‘𝑉)(𝐹‘𝑓)))
118	117	ralrimivva 3172	. 2 ⊢ (𝜑 → ∀ℎ ∈ 𝑆 ∀𝑓 ∈ 𝑇 ((𝐸‘ℎ)(le‘𝑊)𝑓 ↔ ℎ(le‘𝑉)(𝐹‘𝑓)))
119	112	ipobas 18437	. . . 4 ⊢ (𝑆 ∈ V → 𝑆 = (Base‘𝑉))
120	109, 119	ax-mp 5	. . 3 ⊢ 𝑆 = (Base‘𝑉)
121	104	ipobas 18437	. . . 4 ⊢ (𝑇 ∈ V → 𝑇 = (Base‘𝑊))
122	103, 121	ax-mp 5	. . 3 ⊢ 𝑇 = (Base‘𝑊)
123		nsgmgc.j	. . 3 ⊢ 𝐽 = (𝑉MGalConn𝑊)
124	112	ipopos 18442	. . . 4 ⊢ 𝑉 ∈ Poset
125		posprs 18222	. . . 4 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
126	124, 125	mp1i 13	. . 3 ⊢ (𝜑 → 𝑉 ∈ Proset )
127	104	ipopos 18442	. . . 4 ⊢ 𝑊 ∈ Poset
128		posprs 18222	. . . 4 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
129	127, 128	mp1i 13	. . 3 ⊢ (𝜑 → 𝑊 ∈ Proset )
130	120, 122, 113, 105, 123, 126, 129	mgcval 32929	. 2 ⊢ (𝜑 → (𝐸𝐽𝐹 ↔ ((𝐸:𝑆⟶𝑇 ∧ 𝐹:𝑇⟶𝑆) ∧ ∀ℎ ∈ 𝑆 ∀𝑓 ∈ 𝑇 ((𝐸‘ℎ)(le‘𝑊)𝑓 ↔ ℎ(le‘𝑉)(𝐹‘𝑓)))))
131	59, 118, 130	mpbir2and 713	1 ⊢ (𝜑 → 𝐸𝐽𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394  Vcvv 3436   ⊆ wss 3903  {csn 4577   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620   “ cima 5622   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  [cec 8623  Basecbs 17120  lecple 17168   /_s cqus 17409   Proset cproset 18198  Posetcpo 18213  toInccipo 18433  SubGrpcsubg 18999  NrmSGrpcnsg 19000   ~_QG cqg 19001   GrpHom cghm 19091  LSSumclsm 19513  MGalConncmgc 32921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ocomp 17182  df-ds 17183  df-0g 17345  df-imas 17412  df-qus 17413  df-proset 18200  df-poset 18219  df-ipo 18434  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-subg 19002  df-nsg 19003  df-eqg 19004  df-ghm 19092  df-oppg 19225  df-lsm 19515  df-mgc 32923

This theorem is referenced by:  nsgqusf1o  33353