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Theorem grpissubg 16543
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 455 . . . 4  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  S  C_  B
)
21adantl 464 . . 3  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  C_  B
)
3 grpissubg.s . . . . 5  |-  S  =  ( Base `  H
)
43grpbn0 16401 . . . 4  |-  ( H  e.  Grp  ->  S  =/=  (/) )
54ad2antlr 725 . . 3  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  =/=  (/) )
6 ovres 6422 . . . . . . . 8  |-  ( ( a  e.  S  /\  b  e.  S )  ->  ( a ( ( +g  `  G )  |`  ( S  X.  S
) ) b )  =  ( a ( +g  `  G ) b ) )
76adantll 712 . . . . . . 7  |-  ( ( ( ( ( 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 )  |`  ( S  X.  S ) ) b )  =  ( a ( +g  `  G
) b ) )
8 simplr 754 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  H  e.  Grp )
98ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  a  e.  S )  /\  b  e.  S )  ->  H  e.  Grp )
10 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( 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 )
11 simpr 459 . . . . . . . . 9  |-  ( ( ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  a  e.  S )  /\  b  e.  S )  ->  b  e.  S )
12 eqid 2402 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
133, 12grpcl 16385 . . . . . . . . 9  |-  ( ( H  e.  Grp  /\  a  e.  S  /\  b  e.  S )  ->  ( a ( +g  `  H ) b )  e.  S )
149, 10, 11, 13syl3anc 1230 . . . . . . . 8  |-  ( ( ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G
)  |`  ( S  X.  S ) ) ) )  /\  a  e.  S )  /\  b  e.  S )  ->  (
a ( +g  `  H
) b )  e.  S )
15 oveq 6283 . . . . . . . . . . . . 13  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( a ( +g  `  H ) b )  =  ( a ( ( +g  `  G )  |`  ( S  X.  S ) ) b ) )
1615eqcomd 2410 . . . . . . . . . . . 12  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( a ( ( +g  `  G
)  |`  ( S  X.  S ) ) b )  =  ( a ( +g  `  H
) b ) )
1716eleq1d 2471 . . . . . . . . . . 11  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( ( a ( ( +g  `  G
)  |`  ( S  X.  S ) ) b )  e.  S  <->  ( a
( +g  `  H ) b )  e.  S
) )
1817adantl 464 . . . . . . . . . 10  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  ( (
a ( ( +g  `  G )  |`  ( S  X.  S ) ) b )  e.  S  <->  ( a ( +g  `  H
) b )  e.  S ) )
1918adantl 464 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  ( ( a ( ( +g  `  G
)  |`  ( S  X.  S ) ) b )  e.  S  <->  ( a
( +g  `  H ) b )  e.  S
) )
2019ad2antrr 724 . . . . . . . 8  |-  ( ( ( ( ( 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 )  |`  ( S  X.  S
) ) b )  e.  S  <->  ( a
( +g  `  H ) b )  e.  S
) )
2114, 20mpbird 232 . . . . . . 7  |-  ( ( ( ( ( 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 )  |`  ( S  X.  S ) ) b )  e.  S
)
227, 21eqeltrrd 2491 . . . . . 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 )
2322ralrimiva 2817 . . . . 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
)
24 simpl 455 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  H  e.  Grp )  ->  G  e.  Grp )
2524adantr 463 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  G  e.  Grp )
26 grpissubg.b . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
2726sseq2i 3466 . . . . . . . . . . 11  |-  ( S 
C_  B  <->  S  C_  ( Base `  G ) )
2827biimpi 194 . . . . . . . . . 10  |-  ( S 
C_  B  ->  S  C_  ( Base `  G
) )
2928adantr 463 . . . . . . . . 9  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  S  C_  ( Base `  G ) )
3029adantl 464 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  C_  ( Base `  G ) )
31 ovres 6422 . . . . . . . . . . 11  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x ( ( +g  `  G )  |`  ( S  X.  S
) ) y )  =  ( x ( +g  `  G ) y ) )
3231adantl 464 . . . . . . . . . 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 ) )
33 oveq 6283 . . . . . . . . . . . . 13  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( x ( +g  `  H ) y )  =  ( x ( ( +g  `  G )  |`  ( S  X.  S ) ) y ) )
3433adantl 464 . . . . . . . . . . . 12  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  ( x
( +g  `  H ) y )  =  ( x ( ( +g  `  G )  |`  ( S  X.  S ) ) y ) )
3534eqcomd 2410 . . . . . . . . . . 11  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  ( x
( ( +g  `  G
)  |`  ( S  X.  S ) ) y )  =  ( x ( +g  `  H
) y ) )
3635ad2antlr 725 . . . . . . . . . 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 ) )
3732, 36eqtr3d 2445 . . . . . . . . 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 ) )
3837ralrimivva 2824 . . . . . . . 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 ) )
3925, 8, 3, 30, 38grpinvssd 16437 . . . . . . 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
) ) )
4039imp 427 . . . . . 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 )
)
41 eqid 2402 . . . . . . . . . 10  |-  ( invg `  H )  =  ( invg `  H )
423, 41grpinvcl 16417 . . . . . . . . 9  |-  ( ( H  e.  Grp  /\  a  e.  S )  ->  ( ( invg `  H ) `  a
)  e.  S )
4342ex 432 . . . . . . . 8  |-  ( H  e.  Grp  ->  (
a  e.  S  -> 
( ( invg `  H ) `  a
)  e.  S ) )
4443ad2antlr 725 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  ( a  e.  S  ->  ( ( invg `  H ) `
 a )  e.  S ) )
4544imp 427 . . . . . 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 )
4640, 45eqeltrrd 2491 . . . . 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 )
4723, 46jca 530 . . . 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 ) )
4847ralrimiva 2817 . . 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 2402 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
50 eqid 2402 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
5126, 49, 50issubg2 16538 . . . 4  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( S  C_  B  /\  S  =/=  (/)  /\  A. a  e.  S  ( A. b  e.  S  ( a ( +g  `  G ) b )  e.  S  /\  (
( invg `  G ) `  a
)  e.  S ) ) ) )
5251ad2antrr 724 . . 3  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  ( S  e.  (SubGrp `  G )  <->  ( S  C_  B  /\  S  =/=  (/)  /\  A. a  e.  S  ( A. b  e.  S  (
a ( +g  `  G
) b )  e.  S  /\  ( ( invg `  G
) `  a )  e.  S ) ) ) )
532, 5, 48, 52mpbir3and 1180 . 2  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  e.  (SubGrp `  G ) )
5453ex 432 1  |-  ( ( G  e.  Grp  /\  H  e.  Grp )  ->  ( ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) )  ->  S  e.  (SubGrp `  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753    C_ wss 3413   (/)c0 3737    X. cxp 4820    |` cres 4824   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   Grpcgrp 16375   invgcminusg 16376  SubGrpcsubg 16517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-subg 16520
This theorem is referenced by:  resgrpisgrp  16544  pgrpsubgsymg  16755
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