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Theorem subgacs 17830

Description: Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

**Hypothesis**
Ref	Expression
subgacs.b	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
subgacs	⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))

Proof of Theorem subgacs Dummy variables 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . 6 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
2	1	issubg3 17813	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑠 ∈ (SubGrp‘𝐺) ↔ (𝑠 ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ 𝑠 ((inv_g‘𝐺)‘𝑥) ∈ 𝑠)))
3		subgacs.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
4	3	submss 17551	. . . . . . . . 9 ⊢ (𝑠 ∈ (SubMnd‘𝐺) → 𝑠 ⊆ 𝐵)
5	4	adantl 473	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (SubMnd‘𝐺)) → 𝑠 ⊆ 𝐵)
6		selpw 4309	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐵 ↔ 𝑠 ⊆ 𝐵)
7	5, 6	sylibr 224	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (SubMnd‘𝐺)) → 𝑠 ∈ 𝒫 𝐵)
8		eleq2w 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝑠 → (((inv_g‘𝐺)‘𝑥) ∈ 𝑦 ↔ ((inv_g‘𝐺)‘𝑥) ∈ 𝑠))
9	8	raleqbi1dv 3285	. . . . . . . 8 ⊢ (𝑦 = 𝑠 → (∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝑠 ((inv_g‘𝐺)‘𝑥) ∈ 𝑠))
10	9	elrab3 3505	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝐵 → (𝑠 ∈ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦} ↔ ∀𝑥 ∈ 𝑠 ((inv_g‘𝐺)‘𝑥) ∈ 𝑠))
11	7, 10	syl 17	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (SubMnd‘𝐺)) → (𝑠 ∈ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦} ↔ ∀𝑥 ∈ 𝑠 ((inv_g‘𝐺)‘𝑥) ∈ 𝑠))
12	11	pm5.32da 676	. . . . 5 ⊢ (𝐺 ∈ Grp → ((𝑠 ∈ (SubMnd‘𝐺) ∧ 𝑠 ∈ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦}) ↔ (𝑠 ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ 𝑠 ((inv_g‘𝐺)‘𝑥) ∈ 𝑠)))
13	2, 12	bitr4d 271	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑠 ∈ (SubGrp‘𝐺) ↔ (𝑠 ∈ (SubMnd‘𝐺) ∧ 𝑠 ∈ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦})))
14		elin 3939	. . . 4 ⊢ (𝑠 ∈ ((SubMnd‘𝐺) ∩ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦}) ↔ (𝑠 ∈ (SubMnd‘𝐺) ∧ 𝑠 ∈ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦}))
15	13, 14	syl6bbr 278	. . 3 ⊢ (𝐺 ∈ Grp → (𝑠 ∈ (SubGrp‘𝐺) ↔ 𝑠 ∈ ((SubMnd‘𝐺) ∩ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦})))
16	15	eqrdv 2758	. 2 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = ((SubMnd‘𝐺) ∩ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦}))
17		fvex 6362	. . . . 5 ⊢ (Base‘𝐺) ∈ V
18	3, 17	eqeltri 2835	. . . 4 ⊢ 𝐵 ∈ V
19		mreacs 16520	. . . 4 ⊢ (𝐵 ∈ V → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵))
20	18, 19	mp1i 13	. . 3 ⊢ (𝐺 ∈ Grp → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵))
21		grpmnd 17630	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
22	3	submacs 17566	. . . 4 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
23	21, 22	syl 17	. . 3 ⊢ (𝐺 ∈ Grp → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
24	3, 1	grpinvcl 17668	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((inv_g‘𝐺)‘𝑥) ∈ 𝐵)
25	24	ralrimiva 3104	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ((inv_g‘𝐺)‘𝑥) ∈ 𝐵)
26		acsfn1 16523	. . . 4 ⊢ ((𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐵 ((inv_g‘𝐺)‘𝑥) ∈ 𝐵) → {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦} ∈ (ACS‘𝐵))
27	18, 25, 26	sylancr 698	. . 3 ⊢ (𝐺 ∈ Grp → {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦} ∈ (ACS‘𝐵))
28		mreincl 16461	. . 3 ⊢ (((ACS‘𝐵) ∈ (Moore‘𝒫 𝐵) ∧ (SubMnd‘𝐺) ∈ (ACS‘𝐵) ∧ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦} ∈ (ACS‘𝐵)) → ((SubMnd‘𝐺) ∩ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦}) ∈ (ACS‘𝐵))
29	20, 23, 27, 28	syl3anc 1477	. 2 ⊢ (𝐺 ∈ Grp → ((SubMnd‘𝐺) ∩ {𝑦 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝑦 ((inv_g‘𝐺)‘𝑥) ∈ 𝑦}) ∈ (ACS‘𝐵))
30	16, 29	eqeltrd 2839	1 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 {crab 3054 Vcvv 3340 ∩ cin 3714 ⊆ wss 3715 𝒫 cpw 4302 ‘cfv 6049 Basecbs 16059 Moorecmre 16444 ACScacs 16447 Mndcmnd 17495 SubMndcsubmnd 17535 Grpcgrp 17623 inv_gcminusg 17624 SubGrpcsubg 17789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-0g 16304 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-grp 17626 df-minusg 17627 df-subg 17792

This theorem is referenced by: nsgacs 17831 cycsubg2 17832 cycsubg2cl 17833 odf1o1 18187 lsmmod 18288 dmdprdd 18598 dprdfeq0 18621 dprdspan 18626 dprdres 18627 dprdss 18628 dprdz 18629 subgdmdprd 18633 subgdprd 18634 dprdsn 18635 dprd2dlem1 18640 dprd2da 18641 dmdprdsplit2lem 18644 ablfac1b 18669 pgpfac1lem1 18673 pgpfac1lem2 18674 pgpfac1lem3a 18675 pgpfac1lem3 18676 pgpfac1lem4 18677 pgpfac1lem5 18678 pgpfaclem1 18680 pgpfaclem2 18681 lssacs 19169 subrgacs 38272 proot1mul 38279 proot1hash 38280

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