MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinveu Structured version   Unicode version

Theorem grpoinveu 24886
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 24882 . . . 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 457 . . . . . 6  |-  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
54reximi 2925 . . . . 5  |-  ( E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  E. y  e.  X  ( y G A )  =  U )
65adantl 466 . . . 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 2488 . . . . . . . . . . . 12  |-  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  ( y G A )  =  ( z G A ) )
91grporcan 24885 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
y G A )  =  ( z G A )  <->  y  =  z ) )
108, 9syl5ib 219 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  (
y  e.  X  /\  z  e.  X  /\  A  e.  X )
)  ->  ( (
( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
11103exp2 1209 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  ( z  e.  X  ->  ( A  e.  X  ->  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) ) ) ) )
1211com24 87 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( z  e.  X  ->  (
y  e.  X  -> 
( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) ) ) ) )
1312imp41 593 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  z  e.  X )  /\  y  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1413an32s 802 . . . . . . 7  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  y  =  z ) )
1514expd 436 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  z  e.  X )  ->  (
( y G A )  =  U  -> 
( ( z G A )  =  U  ->  y  =  z ) ) )
1615ralrimdva 2875 . . . . 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 553 . . . 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 2931 . . 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 6282 . . . 4  |-  ( y  =  z  ->  (
y G A )  =  ( z G A ) )
2120eqeq1d 2462 . . 3  |-  ( y  =  z  ->  (
( y G A )  =  U  <->  ( z G A )  =  U ) )
2221reu8 3292 . 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 212 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   E!wreu 2809   ran crn 4993   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24850  GIdcgi 24851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-riota 6236  df-ov 6278  df-grpo 24855  df-gid 24856
This theorem is referenced by:  grpoinvcl  24890  grpoinv  24891
  Copyright terms: Public domain W3C validator