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Theorem grpoinveu 20905
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 20901 . . . 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 443 . . . . . 6  |-  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
54reximi 2663 . . . . 5  |-  ( E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  E. y  e.  X  ( y G A )  =  U )
65adantl 452 . . . 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 15 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E. y  e.  X  ( y G A )  =  U )
8 eqtr3 2315 . . . . . . . . . . . 12  |-  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  ( y G A )  =  ( z G A ) )
91grporcan 20904 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
y G A )  =  ( z G A )  <->  y  =  z ) )
108, 9syl5ib 210 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
11103exp2 1169 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  ( z  e.  X  ->  ( A  e.  X  ->  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) ) ) ) )
1211com24 81 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( z  e.  X  ->  (
y  e.  X  -> 
( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) ) ) ) )
1312imp41 576 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  z  e.  X )  /\  y  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1413an32s 779 . . . . . . 7  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1514exp3a 425 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( y G A )  =  U  -> 
( ( z G A )  =  U  ->  y  =  z ) ) )
1615ralrimdva 2646 . . . . 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 536 . . . 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 2668 . . 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 14 . 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 5881 . . . 4  |-  ( y  =  z  ->  (
y G A )  =  ( z G A ) )
2120eqeq1d 2304 . . 3  |-  ( y  =  z  ->  (
( y G A )  =  U  <->  ( z G A )  =  U ) )
2221reu8 2974 . 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 203 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870
This theorem is referenced by:  grpoinvcl  20909  grpoinv  20910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875
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