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Theorem grpoinveu 20882
Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinveu.1  |-  X  =  ran  G
grpinveu.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoinveu  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
Distinct variable groups:    y, A    y, G    y, U    y, X
Dummy variable  z is distinct from all other variables.

Proof of Theorem grpoinveu
StepHypRef Expression
1 grpinveu.1 . . . . 5  |-  X  =  ran  G
2 grpinveu.2 . . . . 5  |-  U  =  (GId `  G )
31, 2grpoidinv2 20878 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
4 simpl 445 . . . . . 6  |-  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
54reximi 2652 . . . . 5  |-  ( E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  E. y  e.  X  ( y G A )  =  U )
65adantl 454 . . . 4  |-  ( ( ( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) )  ->  E. y  e.  X  ( y G A )  =  U )
73, 6syl 17 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E. y  e.  X  ( y G A )  =  U )
8 eqtr3 2304 . . . . . . . . . . . 12  |-  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  ( y G A )  =  ( z G A ) )
91grporcan 20881 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
y G A )  =  ( z G A )  <->  y  =  z ) )
108, 9syl5ib 212 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
11103exp2 1171 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  ( z  e.  X  ->  ( A  e.  X  ->  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) ) ) ) )
1211com24 83 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( z  e.  X  ->  (
y  e.  X  -> 
( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) ) ) ) )
1312imp41 578 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  z  e.  X )  /\  y  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1413an32s 781 . . . . . . 7  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1514exp3a 427 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( y G A )  =  U  -> 
( ( z G A )  =  U  ->  y  =  z ) ) )
1615ralrimdva 2635 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( (
y G A )  =  U  ->  A. z  e.  X  ( (
z G A )  =  U  ->  y  =  z ) ) )
1716ancld 538 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( (
y G A )  =  U  ->  (
( y G A )  =  U  /\  A. z  e.  X  ( ( z G A )  =  U  -> 
y  =  z ) ) ) )
1817reximdva 2657 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( E. y  e.  X  ( y G A )  =  U  ->  E. y  e.  X  ( ( y G A )  =  U  /\  A. z  e.  X  ( ( z G A )  =  U  ->  y  =  z ) ) ) )
197, 18mpd 16 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E. y  e.  X  ( (
y G A )  =  U  /\  A. z  e.  X  (
( z G A )  =  U  -> 
y  =  z ) ) )
20 oveq1 5827 . . . 4  |-  ( y  =  z  ->  (
y G A )  =  ( z G A ) )
2120eqeq1d 2293 . . 3  |-  ( y  =  z  ->  (
( y G A )  =  U  <->  ( z G A )  =  U ) )
2221reu8 2963 . 2  |-  ( E! y  e.  X  ( y G A )  =  U  <->  E. y  e.  X  ( (
y G A )  =  U  /\  A. z  e.  X  (
( z G A )  =  U  -> 
y  =  z ) ) )
2319, 22sylibr 205 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   A.wral 2545   E.wrex 2546   E!wreu 2547   ran crn 4690   ` cfv 5222  (class class class)co 5820   GrpOpcgr 20846  GIdcgi 20847
This theorem is referenced by:  grpoinvcl  20886  grpoinv  20887
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-fo 5228  df-fv 5230  df-ov 5823  df-iota 6253  df-riota 6300  df-grpo 20851  df-gid 20852
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