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Theorem grpidlcan 15590
Description: If left 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
grpidlcan  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( Z  .+  X )  =  X  <-> 
Z  =  .0.  )
)

Proof of Theorem grpidlcan
StepHypRef Expression
1 grpidrcan.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpidrcan.p . . . . 5  |-  .+  =  ( +g  `  G )
3 grpidrcan.o . . . . 5  |-  .0.  =  ( 0g `  G )
41, 2, 3grplid 15566 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
543adant3 1008 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  (  .0.  .+  X
)  =  X )
65eqeq2d 2452 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( Z  .+  X )  =  (  .0.  .+  X )  <->  ( Z  .+  X )  =  X ) )
7 simp1 988 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  G  e.  Grp )
8 simp3 990 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  Z  e.  B )
91, 3grpidcl 15564 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
1093ad2ant1 1009 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  .0.  e.  B )
11 simp2 989 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  X  e.  B )
121, 2grprcan 15569 . . 3  |-  ( ( G  e.  Grp  /\  ( Z  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( Z  .+  X )  =  (  .0.  .+  X
)  <->  Z  =  .0.  ) )
137, 8, 10, 11, 12syl13anc 1220 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( Z  .+  X )  =  (  .0.  .+  X )  <->  Z  =  .0.  ) )
146, 13bitr3d 255 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( Z  .+  X )  =  X  <-> 
Z  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   Basecbs 14172   +g cplusg 14236   0gc0g 14376   Grpcgrp 15408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-riota 6050  df-ov 6092  df-0g 14378  df-mnd 15413  df-grp 15543
This theorem is referenced by:  grpidssd  15600
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