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Theorem grpoidinvlem4 20874
Description: Lemma for grpoidinv 20875. (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 730 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  G  e.  GrpOp
)
2 simplr 731 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  A  e.  X )
3 simpr 447 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  e.  X )
4 grpfo.1 . . . . . . . . 9  |-  X  =  ran  G
54grpoass 20870 . . . . . . . 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 1184 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( ( A G y ) G A )  =  ( A G ( y G A ) ) )
7 oveq2 5866 . . . . . . 7  |-  ( ( y G A )  =  U  ->  ( A G ( y G A ) )  =  ( A G U ) )
86, 7sylan9eq 2335 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
y G A )  =  U )  -> 
( ( A G y ) G A )  =  ( A G U ) )
9 oveq1 5865 . . . . . 6  |-  ( ( A G y )  =  U  ->  (
( A G y ) G A )  =  ( U G A ) )
108, 9sylan9req 2336 . . . . 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 628 . . . 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 587 . . 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 2666 . 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 418 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 358    = wceq 1623    e. wcel 1684   E.wrex 2544   ran crn 4690  (class class class)co 5858   GrpOpcgr 20853
This theorem is referenced by:  grpoidinv  20875  grpoideu  20876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
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