Theorem grpissubg 13105

Description: If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)

Hypotheses

Ref	Expression
grpissubg.b	|- B = ( Base ` G )
grpissubg.s	|- S = ( Base ` H )

Assertion

Ref	Expression
grpissubg	|- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubGrp ` G ) ) )

Proof of Theorem grpissubg

Dummy variables a b x y w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S C_ B )
2	1	adantl 277	. . 3 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S C_ B )
3		grpissubg.s	. . . . . 6 |- S = ( Base ` H )
4		eqid 2189	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
5	3, 4	grpidcl 12945	. . . . 5 |- ( H e. Grp -> ( 0g ` H ) e. S )
6		elex2 2768	. . . . 5 |- ( ( 0g ` H ) e. S -> E. w w e. S )
7	5, 6	syl 14	. . . 4 |- ( H e. Grp -> E. w w e. S )
8	7	ad2antlr 489	. . 3 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> E. w w e. S )
9		grpmnd 12924	. . . . . . . . . . 11 |- ( G e. Grp -> G e. Mnd )
10		mndmgm 12855	. . . . . . . . . . 11 |- ( G e. Mnd -> G e. Mgm )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( G e. Grp -> G e. Mgm )
12		grpmnd 12924	. . . . . . . . . . 11 |- ( H e. Grp -> H e. Mnd )
13		mndmgm 12855	. . . . . . . . . . 11 |- ( H e. Mnd -> H e. Mgm )
14	12, 13	syl 14	. . . . . . . . . 10 |- ( H e. Grp -> H e. Mgm )
15	11, 14	anim12i 338	. . . . . . . . 9 |- ( ( G e. Grp /\ H e. Grp ) -> ( G e. Mgm /\ H e. Mgm ) )
16	15	adantr 276	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( G e. Mgm /\ H e. Mgm ) )
17	16	ad2antrr 488	. . . . . . 7 |- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( G e. Mgm /\ H e. Mgm ) )
18		simpr 110	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) )
19	18	ad2antrr 488	. . . . . . 7 |- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) )
20		simpr 110	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> a e. S )
21	20	anim1i 340	. . . . . . 7 |- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( a e. S /\ b e. S ) )
22		grpissubg.b	. . . . . . . 8 |- B = ( Base ` G )
23	22, 3	mgmsscl 12809	. . . . . . 7 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( a ( +g ` G ) b ) e. S )
24	17, 19, 21, 23	syl3anc 1249	. . . . . 6 |- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( a ( +g ` G ) b ) e. S )
25	24	ralrimiva 2563	. . . . 5 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> A. b e. S ( a ( +g ` G ) b ) e. S )
26		simpl 109	. . . . . . . . 9 |- ( ( G e. Grp /\ H e. Grp ) -> G e. Grp )
27	26	adantr 276	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> G e. Grp )
28		simplr 528	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> H e. Grp )
29	22	sseq2i 3197	. . . . . . . . . . 11 |- ( S C_ B <-> S C_ ( Base ` G ) )
30	29	biimpi 120	. . . . . . . . . 10 |- ( S C_ B -> S C_ ( Base ` G ) )
31	30	adantr 276	. . . . . . . . 9 |- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S C_ ( Base ` G ) )
32	31	adantl 277	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S C_ ( Base ` G ) )
33		ovres 6031	. . . . . . . . . . 11 |- ( ( x e. S /\ y e. S ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` G ) y ) )
34	33	adantl 277	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` G ) y ) )
35		oveq 5897	. . . . . . . . . . . . 13 |- ( ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) -> ( x ( +g ` H ) y ) = ( x ( ( +g ` G ) |` ( S X. S ) ) y ) )
36	35	adantl 277	. . . . . . . . . . . 12 |- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( x ( +g ` H ) y ) = ( x ( ( +g ` G ) |` ( S X. S ) ) y ) )
37	36	eqcomd 2195	. . . . . . . . . . 11 |- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` H ) y ) )
38	37	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` H ) y ) )
39	34, 38	eqtr3d 2224	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
40	39	ralrimivva 2572	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> A. x e. S A. y e. S ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
41	27, 28, 3, 32, 40	grpinvssd 12993	. . . . . . 7 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( a e. S -> ( ( invg ` H ) ` a ) = ( ( invg ` G ) ` a ) ) )
42	41	imp 124	. . . . . 6 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( ( invg ` H ) ` a ) = ( ( invg ` G ) ` a ) )
43		eqid 2189	. . . . . . . 8 |- ( invg ` H ) = ( invg ` H )
44	3, 43	grpinvcl 12964	. . . . . . 7 |- ( ( H e. Grp /\ a e. S ) -> ( ( invg ` H ) ` a ) e. S )
45	44	ad4ant24 516	. . . . . 6 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( ( invg ` H ) ` a ) e. S )
46	42, 45	eqeltrrd 2267	. . . . 5 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( ( invg ` G ) ` a ) e. S )
47	25, 46	jca 306	. . . 4 |- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) )
48	47	ralrimiva 2563	. . 3 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> A. a e. S ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) )
49		eqid 2189	. . . . 5 |- ( +g ` G ) = ( +g ` G )
50		eqid 2189	. . . . 5 |- ( invg ` G ) = ( invg ` G )
51	22, 49, 50	issubg2m 13100	. . . 4 |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ E. w w e. S /\ A. a e. S ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) ) ) )
52	51	ad2antrr 488	. . 3 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ E. w w e. S /\ A. a e. S ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) ) ) )
53	2, 8, 48, 52	mpbir3and 1182	. 2 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S e. (SubGrp ` G ) )
54	53	ex 115	1 |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubGrp ` G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   A.wral 2468   C_ wss 3144   X. cxp 4639   |` cres 4643   ` cfv 5231  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   0gc0g 12733  Mgmcmgm 12802   Mndcmnd 12849   Grpcgrp 12917   invgcminusg 12918  SubGrpcsubg 13078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-ltadd 7946

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-iress 12494  df-plusg 12574  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921  df-subg 13081

This theorem is referenced by:  resgrpisgrp  13106