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Theorem grpissubg 16013
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 15877 . . . . 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 6424 . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
133, 12grpcl 15861 . . . . . . . . . 10  |-  ( ( H  e.  Grp  /\  a  e.  S  /\  b  e.  S )  ->  ( a ( +g  `  H ) b )  e.  S )
149, 10, 11, 13syl3anc 1228 . . . . . . . . 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 6288 . . . . . . . . . . . . . 14  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( a ( +g  `  H ) b )  =  ( a ( ( +g  `  G )  |`  ( S  X.  S ) ) b ) )
1615eqcomd 2475 . . . . . . . . . . . . 13  |-  ( ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) )  ->  ( a ( ( +g  `  G
)  |`  ( S  X.  S ) ) b )  =  ( a ( +g  `  H
) b ) )
1716eleq1d 2536 . . . . . . . . . . . 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 2556 . . . . . . 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 2878 . . . . . 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 3529 . . . . . . . . . . . 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 6424 . . . . . . . . . . . 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 6288 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 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 2510 . . . . . . . . . 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 2885 . . . . . . . . 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 15913 . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( invg `  H )  =  ( invg `  H )
423, 41grpinvcl 15893 . . . . . . . . . 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 2556 . . . . . 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 2878 . . . 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 1176 . . 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 2467 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
51 eqid 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
5226, 50, 51issubg2 16008 . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785    X. cxp 4997    |` cres 5001   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   Grpcgrp 15720   invgcminusg 15721  SubGrpcsubg 15987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-subg 15990
This theorem is referenced by:  resgrpisgrp  16014  pgrpsubgsymg  16225
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