Theorem grpissubg 13085

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 2187	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
5	3, 4	grpidcl 12925	. . . . 5 |- ( H e. Grp -> ( 0g ` H ) e. S )
6		elex2 2765	. . . . 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 12905	. . . . . . . . . . 11 |- ( G e. Grp -> G e. Mnd )
10		mndmgm 12844	. . . . . . . . . . 11 |- ( G e. Mnd -> G e. Mgm )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( G e. Grp -> G e. Mgm )
12		grpmnd 12905	. . . . . . . . . . 11 |- ( H e. Grp -> H e. Mnd )
13		mndmgm 12844	. . . . . . . . . . 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 12798	. . . . . . 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 1248	. . . . . 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 2560	. . . . 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 3194	. . . . . . . . . . 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 6027	. . . . . . . . . . 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 5894	. . . . . . . . . . . . 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 2193	. . . . . . . . . . 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 2222	. . . . . . . . 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 2569	. . . . . . . 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 12973	. . . . . . 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 2187	. . . . . . . 8 |- ( invg ` H ) = ( invg ` H )
44	3, 43	grpinvcl 12944	. . . . . . 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 2265	. . . . 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 2560	. . 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 2187	. . . . 5 |- ( +g ` G ) = ( +g ` G )
50		eqid 2187	. . . . 5 |- ( invg ` G ) = ( invg ` G )
51	22, 49, 50	issubg2m 13080	. . . 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 1181	. 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 979   = wceq 1363   E.wex 1502   e. wcel 2158   A.wral 2465   C_ wss 3141   X. cxp 4636   |` cres 4640   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722  Mgmcmgm 12791   Mndcmnd 12838   Grpcgrp 12898   invgcminusg 12899  SubGrpcsubg 13058

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902  df-subg 13061

This theorem is referenced by:  resgrpisgrp  13086