Theorem subgtgp 21909

Description: A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypothesis

Ref	Expression
subgtgp.h	|- H = ( G ↾s S )

Assertion

Ref	Expression
subgtgp	|- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> H e. TopGrp )

Proof of Theorem subgtgp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgtgp.h	. . . 4 |- H = ( G ↾s S )
2	1	subggrp 17597	. . 3 |- ( S e. (SubGrp ` G ) -> H e. Grp )
3	2	adantl 482	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> H e. Grp )
4		tgptmd 21883	. . 3 |- ( G e. TopGrp -> G e. TopMnd )
5		subgsubm 17616	. . 3 |- ( S e. (SubGrp ` G ) -> S e. (SubMnd ` G ) )
6	1	submtmd 21908	. . 3 |- ( ( G e. TopMnd /\ S e. (SubMnd ` G ) ) -> H e. TopMnd )
7	4, 5, 6	syl2an 494	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> H e. TopMnd )
8	1	subgbas 17598	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
9	8	adantl 482	. . . . . 6 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> S = ( Base ` H ) )
10	9	mpteq1d 4738	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( x e. S |-> ( ( invg ` H ) ` x ) ) = ( x e. ( Base ` H ) |-> ( ( invg ` H ) ` x ) ) )
11		eqid 2622	. . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
12		eqid 2622	. . . . . . . 8 |- ( invg ` H ) = ( invg ` H )
13	1, 11, 12	subginv 17601	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ x e. S ) -> ( ( invg ` G ) ` x ) = ( ( invg ` H ) ` x ) )
14	13	adantll 750	. . . . . 6 |- ( ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) /\ x e. S ) -> ( ( invg ` G ) ` x ) = ( ( invg ` H ) ` x ) )
15	14	mpteq2dva 4744	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( x e. S |-> ( ( invg ` G ) ` x ) ) = ( x e. S |-> ( ( invg ` H ) ` x ) ) )
16		eqid 2622	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
17	16, 12	grpinvf 17466	. . . . . . 7 |- ( H e. Grp -> ( invg ` H ) : ( Base ` H ) --> ( Base ` H ) )
18	3, 17	syl 17	. . . . . 6 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` H ) : ( Base ` H ) --> ( Base ` H ) )
19	18	feqmptd 6249	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` H ) = ( x e. ( Base ` H ) |-> ( ( invg ` H ) ` x ) ) )
20	10, 15, 19	3eqtr4rd 2667	. . . 4 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` H ) = ( x e. S |-> ( ( invg ` G ) ` x ) ) )
21		eqid 2622	. . . . 5 |- ( ( TopOpen ` G ) ↾t S ) = ( ( TopOpen ` G ) ↾t S )
22		eqid 2622	. . . . . . 7 |- ( TopOpen ` G ) = ( TopOpen ` G )
23		eqid 2622	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
24	22, 23	tgptopon 21886	. . . . . 6 |- ( G e. TopGrp -> ( TopOpen ` G ) e. (TopOn ` ( Base ` G ) ) )
25	24	adantr 481	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( TopOpen ` G ) e. (TopOn ` ( Base ` G ) ) )
26	23	subgss 17595	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
27	26	adantl 482	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> S C_ ( Base ` G ) )
28		tgpgrp 21882	. . . . . . . . 9 |- ( G e. TopGrp -> G e. Grp )
29	28	adantr 481	. . . . . . . 8 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> G e. Grp )
30	23, 11	grpinvf 17466	. . . . . . . 8 |- ( G e. Grp -> ( invg ` G ) : ( Base ` G ) --> ( Base ` G ) )
31	29, 30	syl 17	. . . . . . 7 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` G ) : ( Base ` G ) --> ( Base ` G ) )
32	31	feqmptd 6249	. . . . . 6 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` G ) = ( x e. ( Base ` G ) |-> ( ( invg ` G ) ` x ) ) )
33	22, 11	tgpinv 21889	. . . . . . 7 |- ( G e. TopGrp -> ( invg ` G ) e. ( ( TopOpen ` G ) Cn ( TopOpen ` G ) ) )
34	33	adantr 481	. . . . . 6 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` G ) e. ( ( TopOpen ` G ) Cn ( TopOpen ` G ) ) )
35	32, 34	eqeltrrd 2702	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( x e. ( Base ` G ) |-> ( ( invg ` G ) ` x ) ) e. ( ( TopOpen ` G ) Cn ( TopOpen ` G ) ) )
36	21, 25, 27, 35	cnmpt1res 21479	. . . 4 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( x e. S |-> ( ( invg ` G ) ` x ) ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( TopOpen ` G ) ) )
37	20, 36	eqeltrd 2701	. . 3 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( TopOpen ` G ) ) )
38		frn 6053	. . . . . 6 |- ( ( invg ` H ) : ( Base ` H ) --> ( Base ` H ) -> ran ( invg ` H ) C_ ( Base ` H ) )
39	18, 38	syl 17	. . . . 5 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ran ( invg ` H ) C_ ( Base ` H ) )
40	39, 9	sseqtr4d 3642	. . . 4 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ran ( invg ` H ) C_ S )
41		cnrest2 21090	. . . 4 |- ( ( ( TopOpen ` G ) e. (TopOn ` ( Base ` G ) ) /\ ran ( invg ` H ) C_ S /\ S C_ ( Base ` G ) ) -> ( ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( TopOpen ` G ) ) <-> ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( ( TopOpen ` G ) ↾t S ) ) ) )
42	25, 40, 27, 41	syl3anc 1326	. . 3 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( TopOpen ` G ) ) <-> ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( ( TopOpen ` G ) ↾t S ) ) ) )
43	37, 42	mpbid 222	. 2 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( ( TopOpen ` G ) ↾t S ) ) )
44	1, 22	resstopn 20990	. . 3 |- ( ( TopOpen ` G ) ↾t S ) = ( TopOpen ` H )
45	44, 12	istgp 21881	. 2 |- ( H e. TopGrp <-> ( H e. Grp /\ H e. TopMnd /\ ( invg ` H ) e. ( ( ( TopOpen ` G ) ↾t S ) Cn ( ( TopOpen ` G ) ↾t S ) ) ) )
46	3, 7, 43, 45	syl3anbrc 1246	1 |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> H e. TopGrp )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   ↾_t crest 16081   TopOpenctopn 16082  SubMndcsubmnd 17334   Grpcgrp 17422   invgcminusg 17423  SubGrpcsubg 17588  TopOnctopon 20715   Cn ccn 21028  TopMndctmd 21874   TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-tset 15960  df-rest 16083  df-topn 16084  df-0g 16102  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-tx 21365  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  qqhcn  30035