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Theorem grpoidinvlem1 25407
Description: Lemma for grpoidinv 25411. (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 1285 . . . 4  |-  ( ( Y  e.  X  /\  A  e.  X )  ->  ( Y  e.  X  /\  A  e.  X  /\  A  e.  X
) )
3 grpfo.1 . . . . 5  |-  X  =  ran  G
43grpoass 25406 . . . 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 472 . . 3  |-  ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X )
)  ->  ( ( Y G A ) G A )  =  ( Y G ( A G A ) ) )
65adantr 463 . 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 6277 . . 3  |-  ( ( Y G A )  =  U  ->  (
( Y G A ) G A )  =  ( U G A ) )
87ad2antrl 725 . 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 6278 . . . 4  |-  ( ( A G A )  =  A  ->  ( Y G ( A G A ) )  =  ( Y G A ) )
109ad2antll 726 . . 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 754 . . 3  |-  ( ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X
) )  /\  (
( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( Y G A )  =  U )
1210, 11eqtrd 2495 . 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 2503 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ran crn 4989  (class class class)co 6270   GrpOpcgr 25389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-ov 6273  df-grpo 25394
This theorem is referenced by:  grpoidinvlem3  25409
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