Theorem grpissubg 13324

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 2196	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
5	3, 4	grpidcl 13161	. . . . 5 |- ( H e. Grp -> ( 0g ` H ) e. S )
6		elex2 2779	. . . . 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 13139	. . . . . . . . . . 11 |- ( G e. Grp -> G e. Mnd )
10		mndmgm 13063	. . . . . . . . . . 11 |- ( G e. Mnd -> G e. Mgm )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( G e. Grp -> G e. Mgm )
12		grpmnd 13139	. . . . . . . . . . 11 |- ( H e. Grp -> H e. Mnd )
13		mndmgm 13063	. . . . . . . . . . 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 13004	. . . . . . 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 2570	. . . . 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 3210	. . . . . . . . . . 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 6063	. . . . . . . . . . 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 5928	. . . . . . . . . . . . 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 2202	. . . . . . . . . . 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 2231	. . . . . . . . 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 2579	. . . . . . . 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 13209	. . . . . . 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 2196	. . . . . . . 8 |- ( invg ` H ) = ( invg ` H )
44	3, 43	grpinvcl 13180	. . . . . . 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 2274	. . . . 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 2570	. . 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 2196	. . . . 5 |- ( +g ` G ) = ( +g ` G )
50		eqid 2196	. . . . 5 |- ( invg ` G ) = ( invg ` G )
51	22, 49, 50	issubg2m 13319	. . . 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 1506   e. wcel 2167   A.wral 2475   C_ wss 3157   X. cxp 4661   |` cres 4665   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927  Mgmcmgm 12997   Mndcmnd 13057   Grpcgrp 13132   invgcminusg 13133  SubGrpcsubg 13297

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300

This theorem is referenced by:  resgrpisgrp  13325