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Theorem grpoidinvlem4 21645
Description: Lemma for grpoidinv 21646. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1  |-  X  =  ran  G
Assertion
Ref Expression
grpoidinvlem4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )  -> 
( A G U )  =  ( U G A ) )
Distinct variable groups:    y, A    y, G    y, X    y, U

Proof of Theorem grpoidinvlem4
StepHypRef Expression
1 simpll 731 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  G  e.  GrpOp
)
2 simplr 732 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  A  e.  X )
3 simpr 448 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  e.  X )
4 grpfo.1 . . . . . . . . 9  |-  X  =  ran  G
54grpoass 21641 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  y  e.  X  /\  A  e.  X )
)  ->  ( ( A G y ) G A )  =  ( A G ( y G A ) ) )
61, 2, 3, 2, 5syl13anc 1186 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( ( A G y ) G A )  =  ( A G ( y G A ) ) )
7 oveq2 6030 . . . . . . 7  |-  ( ( y G A )  =  U  ->  ( A G ( y G A ) )  =  ( A G U ) )
86, 7sylan9eq 2441 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
y G A )  =  U )  -> 
( ( A G y ) G A )  =  ( A G U ) )
9 oveq1 6029 . . . . . 6  |-  ( ( A G y )  =  U  ->  (
( A G y ) G A )  =  ( U G A ) )
108, 9sylan9req 2442 . . . . 5  |-  ( ( ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X )  /\  (
y G A )  =  U )  /\  ( A G y )  =  U )  -> 
( A G U )  =  ( U G A ) )
1110anasss 629 . . . 4  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
( y G A )  =  U  /\  ( A G y )  =  U ) )  ->  ( A G U )  =  ( U G A ) )
1211exp31 588 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
y  e.  X  -> 
( ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  ( A G U )  =  ( U G A ) ) ) )
1312rexlimdv 2774 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  ( A G U )  =  ( U G A ) ) )
1413imp 419 1  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )  -> 
( A G U )  =  ( U G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2652   ran crn 4821  (class class class)co 6022   GrpOpcgr 21624
This theorem is referenced by:  grpoidinv  21646  grpoideu  21647
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-ov 6025  df-grpo 21629
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