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Theorem grpinveu 13246
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 13227 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
5 eqtr3 2088 . . . . . . . . . . . 12  |-  ( ( ( y  .+  X
)  =  .0.  /\  ( z  .+  X
)  =  .0.  )  ->  ( y  .+  X
)  =  ( z 
.+  X ) )
61, 2grprcan 13245 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( y  .+  X
)  =  ( z 
.+  X )  <->  y  =  z ) )
75, 6syl5ib 208 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
873exp2 1128 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (
y  e.  B  -> 
( z  e.  B  ->  ( X  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
98com24 81 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( z  e.  B  -> 
( y  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
109imp41 570 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  z  e.  B )  /\  y  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1110an32s 739 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1211exp3a 422 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( y  .+  X
)  =  .0.  ->  ( ( z  .+  X
)  =  .0.  ->  y  =  z ) ) )
1312ralrimdva 2360 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
y  .+  X )  =  .0.  ->  A. z  e.  B  ( (
z  .+  X )  =  .0.  ->  y  =  z ) ) )
1413ancld 530 . . . 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 2382 . . 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 14 . 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 5377 . . . 4  |-  ( y  =  z  ->  (
y  .+  X )  =  ( z  .+  X ) )
1817eqeq1d 2077 . . 3  |-  ( y  =  z  ->  (
( y  .+  X
)  =  .0.  <->  ( z  .+  X )  =  .0.  ) )
1918reu8 2651 . 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 201 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E! y  e.  B  ( y  .+  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 356    /\ w3a 896    = wceq 1524    e. wcel 1526   A.wral 2279   E.wrex 2280   E!wreu 2281   ` cfv 4278  (class class class)co 5370   Basecbs 12272   +g cplusg 12327   0gc0g 12505   Grpcgrp 12934
This theorem is referenced by:  grpinvf  13256  grplinv  13258  isgrpinv  13262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-sep 3711  ax-nul 3719  ax-pr 3779  ax-un 4071
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 898  df-ex 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2286  df-v 2482  df-sbc 2656  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-nul 3078  df-if 3187  df-sn 3266  df-pr 3267  df-op 3269  df-uni 3435  df-br 3597  df-opab 3651  df-mpt 3652  df-id 3870  df-xp 4280  df-rel 4281  df-cnv 4282  df-co 4283  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fun 4288  df-fv 4294  df-ov 5373  df-iota 5780  df-0g 12509  df-mnd 12939  df-grp 13219
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