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

Theorem grpoinveu 23854
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 23850 . . . 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 2922 . . . . 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 2479 . . . . . . . . . . . 12  |-  ( ( ( y G A )  =  U  /\  ( z G A )  =  U )  ->  ( y G A )  =  ( z G A ) )
91grporcan 23853 . . . . . . . . . . . 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 1206 . . . . . . . . . 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 2905 . . . . 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 2927 . . 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 6200 . . . 4  |-  ( y  =  z  ->  (
y G A )  =  ( z G A ) )
2120eqeq1d 2453 . . 3  |-  ( y  =  z  ->  (
( y G A )  =  U  <->  ( z G A )  =  U ) )
2221reu8 3255 . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   E!wreu 2797   ran crn 4942   ` cfv 5519  (class class class)co 6193   GrpOpcgr 23818  GIdcgi 23819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-riota 6154  df-ov 6196  df-grpo 23823  df-gid 23824
This theorem is referenced by:  grpoinvcl  23858  grpoinv  23859
  Copyright terms: Public domain W3C validator