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Theorem grpasscan2 12820
Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
Hypotheses
Ref Expression
grplcan.b  |-  B  =  ( Base `  G
)
grplcan.p  |-  .+  =  ( +g  `  G )
grpasscan1.n  |-  N  =  ( invg `  G )
Assertion
Ref Expression
grpasscan2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( N `  Y ) )  .+  Y )  =  X )

Proof of Theorem grpasscan2
StepHypRef Expression
1 simp1 997 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
2 simp2 998 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 grplcan.b . . . . 5  |-  B  =  ( Base `  G
)
4 grpasscan1.n . . . . 5  |-  N  =  ( invg `  G )
53, 4grpinvcl 12808 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
653adant2 1016 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  Y
)  e.  B )
7 simp3 999 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
8 grplcan.p . . . 4  |-  .+  =  ( +g  `  G )
93, 8grpass 12773 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  ( N `  Y
)  e.  B  /\  Y  e.  B )
)  ->  ( ( X  .+  ( N `  Y ) )  .+  Y )  =  ( X  .+  ( ( N `  Y ) 
.+  Y ) ) )
101, 2, 6, 7, 9syl13anc 1240 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( N `  Y ) )  .+  Y )  =  ( X  .+  ( ( N `  Y )  .+  Y
) ) )
11 eqid 2177 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
123, 8, 11, 4grplinv 12809 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( N `  Y )  .+  Y
)  =  ( 0g
`  G ) )
13123adant2 1016 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  Y )  .+  Y
)  =  ( 0g
`  G ) )
1413oveq2d 5885 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( N `  Y
)  .+  Y )
)  =  ( X 
.+  ( 0g `  G ) ) )
153, 8, 11grprid 12794 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( 0g `  G ) )  =  X )
16153adant3 1017 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  ( 0g `  G ) )  =  X )
1710, 14, 163eqtrd 2214 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( N `  Y ) )  .+  Y )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5212  (class class class)co 5869   Basecbs 12442   +g cplusg 12515   0gc0g 12650   Grpcgrp 12764   invgcminusg 12765
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8906  df-2 8964  df-ndx 12445  df-slot 12446  df-base 12448  df-plusg 12528  df-0g 12652  df-mgm 12664  df-sgrp 12697  df-mnd 12707  df-grp 12767  df-minusg 12768
This theorem is referenced by:  mulgaddcomlem  12891
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