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Theorem grpoidinvlem4 25351
Description: Lemma for grpoidinv 25352. (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 751 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  G  e.  GrpOp
)
2 simplr 753 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  A  e.  X )
3 simpr 459 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  e.  X )
4 grpfo.1 . . . . . . 7  |-  X  =  ran  G
54grpoass 25347 . . . . . 6  |-  ( ( 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 1228 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( ( A G y ) G A )  =  ( A G ( y G A ) ) )
7 oveq2 6226 . . . . 5  |-  ( ( y G A )  =  U  ->  ( A G ( y G A ) )  =  ( A G U ) )
86, 7sylan9eq 2457 . . . 4  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
y G A )  =  U )  -> 
( ( A G y ) G A )  =  ( A G U ) )
9 oveq1 6225 . . . 4  |-  ( ( A G y )  =  U  ->  (
( A G y ) G A )  =  ( U G A ) )
108, 9sylan9req 2458 . . 3  |-  ( ( ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X )  /\  (
y G A )  =  U )  /\  ( A G y )  =  U )  -> 
( A G U )  =  ( U G A ) )
1110anasss 645 . 2  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
( y G A )  =  U  /\  ( A G y )  =  U ) )  ->  ( A G U )  =  ( U G A ) )
1211r19.29an 2940 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   E.wrex 2747   ran crn 4931  (class class class)co 6218   GrpOpcgr 25330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-ov 6221  df-grpo 25335
This theorem is referenced by:  grpoidinv  25352  grpoideu  25353
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