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Theorem grpoinveu 21767
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

Proof of Theorem grpoinveu
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 grpinveu.1 . . . . 5  |-  X  =  ran  G
2 grpinveu.2 . . . . 5  |-  U  =  (GId `  G )
31, 2grpoidinv2 21763 . . . 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 444 . . . . . 6  |-  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
54reximi 2777 . . . . 5  |-  ( E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  E. y  e.  X  ( y G A )  =  U )
65adantl 453 . . . 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 16 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E. y  e.  X  ( y G A )  =  U )
8 eqtr3 2427 . . . . . . . . . . . 12  |-  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  ( y G A )  =  ( z G A ) )
91grporcan 21766 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
y G A )  =  ( z G A )  <->  y  =  z ) )
108, 9syl5ib 211 . . . . . . . . . . 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 577 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  z  e.  X )  /\  y  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1413an32s 780 . . . . . . 7  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1514exp3a 426 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( y G A )  =  U  -> 
( ( z G A )  =  U  ->  y  =  z ) ) )
1615ralrimdva 2760 . . . . 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 537 . . . 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 2782 . . 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 15 . 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 6051 . . . 4  |-  ( y  =  z  ->  (
y G A )  =  ( z G A ) )
2120eqeq1d 2416 . . 3  |-  ( y  =  z  ->  (
( y G A )  =  U  <->  ( z G A )  =  U ) )
2221reu8 3094 . 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 204 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   E!wreu 2672   ran crn 4842   ` cfv 5417  (class class class)co 6044   GrpOpcgr 21731  GIdcgi 21732
This theorem is referenced by:  grpoinvcl  21771  grpoinv  21772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-ov 6047  df-riota 6512  df-grpo 21736  df-gid 21737
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