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Theorem grpoidinvlem4 23847
Description: Lemma for grpoidinv 23848. (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 753 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  G  e.  GrpOp
)
2 simplr 754 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  A  e.  X )
3 simpr 461 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  y  e.  X )
4 grpfo.1 . . . . . . . . 9  |-  X  =  ran  G
54grpoass 23843 . . . . . . . 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 1221 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( ( A G y ) G A )  =  ( A G ( y G A ) ) )
7 oveq2 6209 . . . . . . 7  |-  ( ( y G A )  =  U  ->  ( A G ( y G A ) )  =  ( A G U ) )
86, 7sylan9eq 2515 . . . . . 6  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  y  e.  X )  /\  (
y G A )  =  U )  -> 
( ( A G y ) G A )  =  ( A G U ) )
9 oveq1 6208 . . . . . 6  |-  ( ( A G y )  =  U  ->  (
( A G y ) G A )  =  ( U G A ) )
108, 9sylan9req 2516 . . . . 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 647 . . . 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 604 . . 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 2946 . 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 429 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 369    = wceq 1370    e. wcel 1758   E.wrex 2800   ran crn 4950  (class class class)co 6201   GrpOpcgr 23826
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fo 5533  df-fv 5535  df-ov 6204  df-grpo 23831
This theorem is referenced by:  grpoidinv  23848  grpoideu  23849
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