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Theorem grpidlcan 16230
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 16206 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
543adant3 1016 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  (  .0.  .+  X
)  =  X )
65eqeq2d 2471 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( Z  .+  X )  =  (  .0.  .+  X )  <->  ( Z  .+  X )  =  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 16204 . . . 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, 2grprcan 16209 . . 3  |-  ( ( G  e.  Grp  /\  ( Z  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( Z  .+  X )  =  (  .0.  .+  X
)  <->  Z  =  .0.  ) )
137, 8, 10, 11, 12syl13anc 1230 . 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 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856   Grpcgrp 16179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-riota 6258  df-ov 6299  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183
This theorem is referenced by:  grpidssd  16240
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