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Theorem grpoidinvlem4 20890
Description: Lemma for grpoidinv 20891. (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 20886 . . . . . . . 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 5882 . . . . . . 7  |-  ( ( y G A )  =  U  ->  ( A G ( y G A ) )  =  ( A G U ) )
86, 7sylan9eq 2348 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
y G A )  =  U )  -> 
( ( A G y ) G A )  =  ( A G U ) )
9 oveq1 5881 . . . . . 6  |-  ( ( A G y )  =  U  ->  (
( A G y ) G A )  =  ( U G A ) )
108, 9sylan9req 2349 . . . . 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 2679 . 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 1632    e. wcel 1696   E.wrex 2557   ran crn 4706  (class class class)co 5874   GrpOpcgr 20869
This theorem is referenced by:  grpoidinv  20891  grpoideu  20892
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-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-grpo 20874
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