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Theorem grpinveu 14351
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 14332 . . 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 14350 . . . . . . . . . . . 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 2595 . . . . 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 2617 . . 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 5717 . . . 4  |-  ( y  =  z  ->  (
y  .+  X )  =  ( z  .+  X ) )
1817eqeq1d 2261 . . 3  |-  ( y  =  z  ->  (
( y  .+  X
)  =  .0.  <->  ( z  .+  X )  =  .0.  ) )
1918reu8 2900 . 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 2509   E.wrex 2510   E!wreu 2511   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   0gc0g 13274   Grpcgrp 14197
This theorem is referenced by:  grpinvf  14361  grplinv  14363  isgrpinv  14367
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 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190  df-0g 13278  df-mnd 14202  df-grp 14324
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