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Theorem grpsubadd0sub 15999
Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.)
Hypotheses
Ref Expression
grpsubid.b  |-  B  =  ( Base `  G
)
grpsubid.o  |-  .0.  =  ( 0g `  G )
grpsubid.m  |-  .-  =  ( -g `  G )
grpsubadd0sub.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
grpsubadd0sub  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  (  .0.  .-  Y ) ) )

Proof of Theorem grpsubadd0sub
StepHypRef Expression
1 grpsubid.b . . . 4  |-  B  =  ( Base `  G
)
2 grpsubadd0sub.p . . . 4  |-  .+  =  ( +g  `  G )
3 eqid 2443 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
4 grpsubid.m . . . 4  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 15967 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
653adant1 1015 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
7 grpsubid.o . . . . 5  |-  .0.  =  ( 0g `  G )
81, 4, 3, 7grpinvval2 15995 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  =  (  .0.  .-  Y ) )
983adant2 1016 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  =  (  .0.  .-  Y ) )
109oveq2d 6297 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  =  ( X  .+  (  .0.  .-  Y ) ) )
116, 10eqtrd 2484 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  (  .0.  .-  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   0gc0g 14714   Grpcgrp 15927   invgcminusg 15928   -gcsg 15929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-sbg 15933
This theorem is referenced by:  chfacfscmulgsum  19234  chfacfpmmulgsum  19238
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