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Theorem grpoidinvlem1 23713
Description: Lemma for grpoidinv 23717. (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 22 . . . . 5  |-  ( ( Y  e.  X  /\  A  e.  X  /\  A  e.  X )  ->  ( Y  e.  X  /\  A  e.  X  /\  A  e.  X
) )
213anidm23 1277 . . . 4  |-  ( ( Y  e.  X  /\  A  e.  X )  ->  ( Y  e.  X  /\  A  e.  X  /\  A  e.  X
) )
3 grpfo.1 . . . . 5  |-  X  =  ran  G
43grpoass 23712 . . . 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 474 . . 3  |-  ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X )
)  ->  ( ( Y G A ) G A )  =  ( Y G ( A G A ) ) )
65adantr 465 . 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 6119 . . 3  |-  ( ( Y G A )  =  U  ->  (
( Y G A ) G A )  =  ( U G A ) )
87ad2antrl 727 . 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 6120 . . . 4  |-  ( ( A G A )  =  A  ->  ( Y G ( A G A ) )  =  ( Y G A ) )
109ad2antll 728 . . 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 755 . . 3  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( Y G A )  =  U )
1210, 11eqtrd 2475 . 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 2483 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4862  (class class class)co 6112   GrpOpcgr 23695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fo 5445  df-fv 5447  df-ov 6115  df-grpo 23700
This theorem is referenced by:  grpoidinvlem3  23715
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