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Theorem grpoidinvlem1 21793
Description: Lemma for grpoidinv 21797. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1  |-  X  =  ran  G
Assertion
Ref Expression
grpoidinvlem1  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( U G A )  =  U )

Proof of Theorem grpoidinvlem1
StepHypRef Expression
1 id 21 . . . . 5  |-  ( ( Y  e.  X  /\  A  e.  X  /\  A  e.  X )  ->  ( Y  e.  X  /\  A  e.  X  /\  A  e.  X
) )
213anidm23 1244 . . . 4  |-  ( ( Y  e.  X  /\  A  e.  X )  ->  ( Y  e.  X  /\  A  e.  X  /\  A  e.  X
) )
3 grpfo.1 . . . . 5  |-  X  =  ran  G
43grpoass 21792 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X  /\  A  e.  X )
)  ->  ( ( Y G A ) G A )  =  ( Y G ( A G A ) ) )
52, 4sylan2 462 . . 3  |-  ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X )
)  ->  ( ( Y G A ) G A )  =  ( Y G ( A G A ) ) )
65adantr 453 . 2  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( ( Y G A ) G A )  =  ( Y G ( A G A ) ) )
7 oveq1 6089 . . 3  |-  ( ( Y G A )  =  U  ->  (
( Y G A ) G A )  =  ( U G A ) )
87ad2antrl 710 . 2  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( ( Y G A ) G A )  =  ( U G A ) )
9 oveq2 6090 . . . 4  |-  ( ( A G A )  =  A  ->  ( Y G ( A G A ) )  =  ( Y G A ) )
109ad2antll 711 . . 3  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( Y G ( A G A ) )  =  ( Y G A ) )
11 simprl 734 . . 3  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( Y G A )  =  U )
1210, 11eqtrd 2469 . 2  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( Y G ( A G A ) )  =  U )
136, 8, 123eqtr3d 2477 1  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( U G A )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4880  (class class class)co 6082   GrpOpcgr 21775
This theorem is referenced by:  grpoidinvlem3  21795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463  df-ov 6085  df-grpo 21780
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