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Theorem grpinveu 14313
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 14294 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
5 eqtr3 2272 . . . . . . . . . . . 12  |-  ( ( ( y  .+  X
)  =  .0.  /\  ( z  .+  X
)  =  .0.  )  ->  ( y  .+  X
)  =  ( z 
.+  X ) )
61, 2grprcan 14312 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( y  .+  X
)  =  ( z 
.+  X )  <->  y  =  z ) )
75, 6syl5ib 212 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
873exp2 1174 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (
y  e.  B  -> 
( z  e.  B  ->  ( X  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
98com24 83 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( z  e.  B  -> 
( y  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
109imp41 579 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  z  e.  B )  /\  y  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1110an32s 782 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1211exp3a 427 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( y  .+  X
)  =  .0.  ->  ( ( z  .+  X
)  =  .0.  ->  y  =  z ) ) )
1312ralrimdva 2593 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
y  .+  X )  =  .0.  ->  A. z  e.  B  ( (
z  .+  X )  =  .0.  ->  y  =  z ) ) )
1413ancld 538 . . . 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 2615 . . 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 16 . 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 5714 . . . 4  |-  ( y  =  z  ->  (
y  .+  X )  =  ( z  .+  X ) )
1817eqeq1d 2261 . . 3  |-  ( y  =  z  ->  (
( y  .+  X
)  =  .0.  <->  ( z  .+  X )  =  .0.  ) )
1918reu8 2898 . 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 205 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E! y  e.  B  ( y  .+  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2507   E.wrex 2508   E!wreu 2509   ` cfv 4589  (class class class)co 5707   Basecbs 12984   +g cplusg 13044   0gc0g 13236   Grpcgrp 14159
This theorem is referenced by:  grpinvf  14323  grplinv  14325  isgrpinv  14329
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4035  ax-nul 4043  ax-pr 4105  ax-un 4400
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2511  df-rex 2512  df-reu 2513  df-rab 2514  df-v 2727  df-sbc 2920  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-nul 3360  df-if 3468  df-sn 3547  df-pr 3548  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-mpt 3973  df-id 4199  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fv 4605  df-ov 5710  df-iota 6140  df-riota 6187  df-0g 13240  df-mnd 14164  df-grp 14286
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