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Theorem grpinveu 16778
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
Dummy variable  z is distinct from all other variables.
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 16759 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
5 eqtr3 2492 . . . . . . . . . . . 12  |-  ( ( ( y  .+  X
)  =  .0.  /\  ( z  .+  X
)  =  .0.  )  ->  ( y  .+  X
)  =  ( z 
.+  X ) )
61, 2grprcan 16777 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( y  .+  X
)  =  ( z 
.+  X )  <->  y  =  z ) )
75, 6syl5ib 227 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B  /\  X  e.  B
) )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
873exp2 1251 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (
y  e.  B  -> 
( z  e.  B  ->  ( X  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
98com24 89 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( z  e.  B  -> 
( y  e.  B  ->  ( ( ( y 
.+  X )  =  .0.  /\  ( z 
.+  X )  =  .0.  )  ->  y  =  z ) ) ) ) )
109imp41 604 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  z  e.  B )  /\  y  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1110an32s 821 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( ( y  .+  X )  =  .0. 
/\  ( z  .+  X )  =  .0.  )  ->  y  =  z ) )
1211expd 443 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  B )  /\  y  e.  B )  /\  z  e.  B )  ->  (
( y  .+  X
)  =  .0.  ->  ( ( z  .+  X
)  =  .0.  ->  y  =  z ) ) )
1312ralrimdva 2812 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  y  e.  B
)  ->  ( (
y  .+  X )  =  .0.  ->  A. z  e.  B  ( (
z  .+  X )  =  .0.  ->  y  =  z ) ) )
1413ancld 562 . . . 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 2858 . . 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 6315 . . . 4  |-  ( y  =  z  ->  (
y  .+  X )  =  ( z  .+  X ) )
1817eqeq1d 2473 . . 3  |-  ( y  =  z  ->  (
( y  .+  X
)  =  .0.  <->  ( z  .+  X )  =  .0.  ) )
1918reu8 3222 . 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 217 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E! y  e.  B  ( y  .+  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   E!wreu 2758   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   0gc0g 15416   Grpcgrp 16747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-riota 6270  df-ov 6311  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751
This theorem is referenced by:  grpinvf  16788  grplinv  16790  isgrpinv  16794
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