Theorem grpissubg 13947

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 2234	. . . . . 6 |- ( 0g ` H ) = ( 0g ` H )
5	3, 4	grpidcl 13784	. . . . 5 |- ( H e. Grp -> ( 0g ` H ) e. S )
6		elex2 2832	. . . . 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 13762	. . . . . . . . . . 11 |- ( G e. Grp -> G e. Mnd )
10		mndmgm 13683	. . . . . . . . . . 11 |- ( G e. Mnd -> G e. Mgm )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( G e. Grp -> G e. Mgm )
12		grpmnd 13762	. . . . . . . . . . 11 |- ( H e. Grp -> H e. Mnd )
13		mndmgm 13683	. . . . . . . . . . 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 13624	. . . . . . 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 1274	. . . . . 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 2617	. . . . 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 529	. . . . . . . 8 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> H e. Grp )
29	22	sseq2i 3269	. . . . . . . . . . 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 6202	. . . . . . . . . . 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 6064	. . . . . . . . . . . . 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 2240	. . . . . . . . . . 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 2269	. . . . . . . . 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 2626	. . . . . . . 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 13832	. . . . . . 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 2234	. . . . . . . 8 |- ( invg ` H ) = ( invg ` H )
44	3, 43	grpinvcl 13803	. . . . . . 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 2312	. . . . 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 2617	. . 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 2234	. . . . 5 |- ( +g ` G ) = ( +g ` G )
50		eqid 2234	. . . . 5 |- ( invg ` G ) = ( invg ` G )
51	22, 49, 50	issubg2m 13942	. . . 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 1207	. 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 1005   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   C_ wss 3214   X. cxp 4752   |` cres 4756   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553  Mgmcmgm 13617   Mndcmnd 13677   Grpcgrp 13755   invgcminusg 13756  SubGrpcsubg 13920

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923

This theorem is referenced by:  resgrpisgrp  13948