Theorem resgrpisgrp 13772

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 other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem resgrpisgrp

Step	Hyp	Ref	Expression
1		grpissubg.b	. . . . 5 |- B = ( Base ` G )
2		grpissubg.s	. . . . 5 |- S = ( Base ` H )
3	1, 2	grpissubg 13771	. . . 4 |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubGrp ` G ) ) )
4	3	imp 124	. . 3 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S e. (SubGrp ` G ) )
5		ibar 301	. . . . . 6 |- ( ( G e. Grp /\ S C_ B ) -> ( ( G ↾s S ) e. Grp <-> ( ( G e. Grp /\ S C_ B ) /\ ( G ↾s S ) e. Grp ) ) )
6	5	ad2ant2r 509	. . . . 5 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( ( G ↾s S ) e. Grp <-> ( ( G e. Grp /\ S C_ B ) /\ ( G ↾s S ) e. Grp ) ) )
7		df-3an 1004	. . . . 5 |- ( ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) <-> ( ( G e. Grp /\ S C_ B ) /\ ( G ↾s S ) e. Grp ) )
8	6, 7	bitr4di 198	. . . 4 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( ( G ↾s S ) e. Grp <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) ) )
9	1	issubg 13750	. . . 4 |- ( S e. (SubGrp ` G ) <-> ( G e. Grp /\ S C_ B /\ ( G ↾s S ) e. Grp ) )
10	8, 9	bitr4di 198	. . 3 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( ( G ↾s S ) e. Grp <-> S e. (SubGrp ` G ) ) )
11	4, 10	mpbird 167	. 2 |- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( G ↾s S ) e. Grp )
12	11	ex 115	1 |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( G ↾s S ) e. Grp ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3198   X. cxp 4721   |` cres 4725   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   +g cplusg 13150   Grpcgrp 13573  SubGrpcsubg 13744

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-subg 13747

This theorem is referenced by: (None)