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Theorem grpinveu 14197
Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinveu.b  |-  B  =  ( Base `  G
)
grpinveu.p  |-  .+  =  ( +g  `  G )
grpinveu.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpinveu  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E! y  e.  B  ( y  .+  X
)  =  .0.  )
Distinct variable groups:    y, B    y, G    y,  .+    y,  .0.    y, X

Proof of Theorem grpinveu
StepHypRef Expression
1 grpinveu.b . . . 4  |-  B  =  ( Base `  G
)
2 grpinveu.p . . . 4  |-  .+  =  ( +g  `  G )
3 grpinveu.o . . . 4  |-  .0.  =  ( 0g `  G )
41, 2, 3grpinvex 14178 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
5 eqtr3 2260 . . . . . . . . . . . 12  |-  ( ( ( y  .+  X
)  =  .0.  /\  ( z  .+  X
)  =  .0.  )  ->  ( y  .+  X
)  =  ( z 
.+  X ) )
61, 2grprcan 14196 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( y  .+  X
)  =  ( z 
.+  X )  <->  y  =  z ) )
75, 6syl5ib 209 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
873exp2 1168 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (
y  e.  B  -> 
( z  e.  B  ->  ( X  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
98com24 82 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( z  e.  B  -> 
( y  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
109imp41 573 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  z  e.  B )  /\  y  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1110an32s 776 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1211exp3a 423 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( y  .+  X
)  =  .0.  ->  ( ( z  .+  X
)  =  .0.  ->  y  =  z ) ) )
1312ralrimdva 2581 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
y  .+  X )  =  .0.  ->  A. z  e.  B  ( (
z  .+  X )  =  .0.  ->  y  =  z ) ) )
1413ancld 532 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
y  .+  X )  =  .0.  ->  ( (
y  .+  X )  =  .0.  /\  A. z  e.  B  ( (
z  .+  X )  =  .0.  ->  y  =  z ) ) ) )
1514reximdva 2603 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( E. y  e.  B  ( y  .+  X )  =  .0. 
->  E. y  e.  B  ( ( y  .+  X )  =  .0. 
/\  A. z  e.  B  ( ( z  .+  X )  =  .0. 
->  y  =  z
) ) ) )
164, 15mpd 15 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E. y  e.  B  ( ( y  .+  X )  =  .0. 
/\  A. z  e.  B  ( ( z  .+  X )  =  .0. 
->  y  =  z
) ) )
17 oveq1 5692 . . . 4  |-  ( y  =  z  ->  (
y  .+  X )  =  ( z  .+  X ) )
1817eqeq1d 2249 . . 3  |-  ( y  =  z  ->  (
( y  .+  X
)  =  .0.  <->  ( z  .+  X )  =  .0.  ) )
1918reu8 2885 . 2  |-  ( E! y  e.  B  ( y  .+  X )  =  .0.  <->  E. y  e.  B  ( (
y  .+  X )  =  .0.  /\  A. z  e.  B  ( (
z  .+  X )  =  .0.  ->  y  =  z ) ) )
2016, 19sylibr 202 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E! y  e.  B  ( y  .+  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 5    /\ wa 357    /\ w3a 933    = wceq 1608    e. wcel 1610   A.wral 2495   E.wrex 2496   E!wreu 2497   ` cfv 4571  (class class class)co 5685   Basecbs 12956   +g cplusg 13016   0gc0g 13208   Grpcgrp 14043
This theorem is referenced by:  grpinvf  14207  grplinv  14209  isgrpinv  14213
This theorem was proved from axioms:  ax-1 6  ax-2 7  ax-3 8  ax-mp 9  ax-5 1522  ax-6 1523  ax-7 1524  ax-gen 1525  ax-8 1612  ax-11 1613  ax-13 1614  ax-14 1615  ax-17 1617  ax-12o 1653  ax-10 1667  ax-9 1673  ax-4 1681  ax-16 1915  ax-ext 2222  ax-sep 4017  ax-nul 4025  ax-pr 4087  ax-un 4382
This theorem depends on definitions:  df-bi 176  df-or 358  df-an 359  df-3an 935  df-tru 1309  df-ex 1527  df-nf 1529  df-sb 1872  df-eu 2106  df-mo 2107  df-clab 2228  df-cleq 2234  df-clel 2237  df-nfc 2362  df-ne 2402  df-ral 2499  df-rex 2500  df-reu 2501  df-rab 2502  df-v 2714  df-sbc 2907  df-dif 3061  df-un 3063  df-in 3065  df-ss 3069  df-nul 3343  df-if 3451  df-sn 3530  df-pr 3531  df-op 3533  df-uni 3708  df-br 3901  df-opab 3955  df-mpt 3956  df-id 4181  df-xp 4573  df-rel 4574  df-cnv 4575  df-co 4576  df-dm 4577  df-rn 4578  df-res 4579  df-ima 4580  df-fun 4581  df-fv 4587  df-ov 5688  df-iota 6117  df-riota 6164  df-0g 13212  df-mnd 14048  df-grp 14170
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