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Theorem grpidrcan 15975
Description: If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
Hypotheses
Ref Expression
grpidrcan.b  |-  B  =  ( Base `  G
)
grpidrcan.p  |-  .+  =  ( +g  `  G )
grpidrcan.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpidrcan  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  X  <-> 
Z  =  .0.  )
)

Proof of Theorem grpidrcan
StepHypRef Expression
1 grpidrcan.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpidrcan.p . . . . 5  |-  .+  =  ( +g  `  G )
3 grpidrcan.o . . . . 5  |-  .0.  =  ( 0g `  G )
41, 2, 3grprid 15953 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  .0.  )  =  X )
543adant3 1016 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .+  .0.  )  =  X )
65eqeq2d 2481 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  ( X  .+  .0.  )  <->  ( X  .+  Z )  =  X ) )
7 simp1 996 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  G  e.  Grp )
8 simp3 998 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  Z  e.  B )
91, 3grpidcl 15950 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
1093ad2ant1 1017 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  .0.  e.  B )
11 simp2 997 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  X  e.  B )
121, 2grplcan 15974 . . 3  |-  ( ( G  e.  Grp  /\  ( Z  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X  .+  Z )  =  ( X  .+  .0.  ) 
<->  Z  =  .0.  )
)
137, 8, 10, 11, 12syl13anc 1230 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  ( X  .+  .0.  )  <->  Z  =  .0.  ) )
146, 13bitr3d 255 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  X  <-> 
Z  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930
This theorem is referenced by: (None)
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