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Theorem grpissubg 15821
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 457 . . . . 5  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  S  C_  B
)
21adantl 466 . . . 4  |-  ( ( ( 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
)
43grpbn0 15687 . . . . 5  |-  ( H  e.  Grp  ->  S  =/=  (/) )
54ad2antlr 726 . . . 4  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  =/=  (/) )
6 ovres 6341 . . . . . . . . 9  |-  ( ( a  e.  S  /\  b  e.  S )  ->  ( a ( ( +g  `  G )  |`  ( S  X.  S
) ) b )  =  ( a ( +g  `  G ) b ) )
76adantll 713 . . . . . . . 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 )  =  ( a ( +g  `  G
) b ) )
8 simplr 754 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  H  e.  Grp )
98ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( 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 . . . . . . . . . 10  |-  ( ( ( ( ( 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 461 . . . . . . . . . 10  |-  ( ( ( ( ( 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 2454 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
133, 12grpcl 15671 . . . . . . . . . 10  |-  ( ( H  e.  Grp  /\  a  e.  S  /\  b  e.  S )  ->  ( a ( +g  `  H ) b )  e.  S )
149, 10, 11, 13syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( ( 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 6207 . . . . . . . . . . . . . 14  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( a ( +g  `  H ) b )  =  ( a ( ( +g  `  G )  |`  ( S  X.  S ) ) b ) )
1615eqcomd 2462 . . . . . . . . . . . . 13  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( a ( ( +g  `  G
)  |`  ( S  X.  S ) ) b )  =  ( a ( +g  `  H
) b ) )
1716eleq1d 2523 . . . . . . . . . . . 12  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( ( a ( ( +g  `  G
)  |`  ( S  X.  S ) ) b )  e.  S  <->  ( a
( +g  `  H ) b )  e.  S
) )
1817adantl 466 . . . . . . . . . . 11  |-  ( ( 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 466 . . . . . . . . . 10  |-  ( ( ( 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 725 . . . . . . . . 9  |-  ( ( ( ( ( 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 . . . . . . . 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
)
227, 21eqeltrrd 2543 . . . . . . 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
) b )  e.  S )
2322ralrimiva 2830 . . . . . 6  |-  ( ( ( ( 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 457 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  H  e.  Grp )  ->  G  e.  Grp )
2524adantr 465 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  G  e.  Grp )
26 grpissubg.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
2726sseq2i 3490 . . . . . . . . . . . 12  |-  ( S 
C_  B  <->  S  C_  ( Base `  G ) )
2827biimpi 194 . . . . . . . . . . 11  |-  ( S 
C_  B  ->  S  C_  ( Base `  G
) )
2928adantr 465 . . . . . . . . . 10  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  S  C_  ( Base `  G ) )
3029adantl 466 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  C_  ( Base `  G ) )
31 ovres 6341 . . . . . . . . . . . 12  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x ( ( +g  `  G )  |`  ( S  X.  S
) ) y )  =  ( x ( +g  `  G ) y ) )
3231adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( 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 6207 . . . . . . . . . . . . . 14  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( x ( +g  `  H ) y )  =  ( x ( ( +g  `  G )  |`  ( S  X.  S ) ) y ) )
3433adantl 466 . . . . . . . . . . . . 13  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  ( x
( +g  `  H ) y )  =  ( x ( ( +g  `  G )  |`  ( S  X.  S ) ) y ) )
3534eqcomd 2462 . . . . . . . . . . . 12  |-  ( ( S  C_  B  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  ( x
( ( +g  `  G
)  |`  ( S  X.  S ) ) y )  =  ( x ( +g  `  H
) y ) )
3635ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ( 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 2497 . . . . . . . . . 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 ) y )  =  ( x ( +g  `  H ) y ) )
3837ralrimivva 2914 . . . . . . . . 9  |-  ( ( ( 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 15723 . . . . . . . 8  |-  ( ( ( 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 429 . . . . . . 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 )
)
41 eqid 2454 . . . . . . . . . . 11  |-  ( invg `  H )  =  ( invg `  H )
423, 41grpinvcl 15703 . . . . . . . . . 10  |-  ( ( H  e.  Grp  /\  a  e.  S )  ->  ( ( invg `  H ) `  a
)  e.  S )
4342ex 434 . . . . . . . . 9  |-  ( H  e.  Grp  ->  (
a  e.  S  -> 
( ( invg `  H ) `  a
)  e.  S ) )
4443ad2antlr 726 . . . . . . . 8  |-  ( ( ( 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 429 . . . . . . 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 )
4640, 45eqeltrrd 2543 . . . . . 6  |-  ( ( ( ( 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 532 . . . . 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  /\  (
( invg `  G ) `  a
)  e.  S ) )
4847ralrimiva 2830 . . . 4  |-  ( ( ( 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 ) )
492, 5, 483jca 1168 . . 3  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  ( S  C_  B  /\  S  =/=  (/)  /\  A. a  e.  S  ( A. b  e.  S  ( a ( +g  `  G ) b )  e.  S  /\  (
( invg `  G ) `  a
)  e.  S ) ) )
50 eqid 2454 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
51 eqid 2454 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
5226, 50, 51issubg2 15816 . . . 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 ) ) ) )
5352ad2antrr 725 . . 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 ) ) ) )
5449, 53mpbird 232 . 2  |-  ( ( ( G  e.  Grp  /\  H  e.  Grp )  /\  ( S  C_  B  /\  ( +g  `  H
)  =  ( ( +g  `  G )  |`  ( S  X.  S
) ) ) )  ->  S  e.  (SubGrp `  G ) )
5554ex 434 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    C_ wss 3437   (/)c0 3746    X. cxp 4947    |` cres 4951   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   Grpcgrp 15530   invgcminusg 15531  SubGrpcsubg 15795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-subg 15798
This theorem is referenced by:  resgrpisgrp  15822  pgrpsubgsymg  16033
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