MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovclg Unicode version

Theorem caovclg 6198
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovclg.1  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
Assertion
Ref Expression
caovclg  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Distinct variable groups:    x, y, A    y, B    x, C, y    x, D, y    x, E, y    ph, x, y   
x, F, y
Allowed substitution hint:    B( x)

Proof of Theorem caovclg
StepHypRef Expression
1 caovclg.1 . . 3  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
21ralrimivva 2758 . 2  |-  ( ph  ->  A. x  e.  C  A. y  e.  D  ( x F y )  e.  E )
3 oveq1 6047 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43eleq1d 2470 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  E  <->  ( A F y )  e.  E ) )
5 oveq2 6048 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eleq1d 2470 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  E  <->  ( A F B )  e.  E
) )
74, 6rspc2v 3018 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ( x F y )  e.  E  ->  ( A F B )  e.  E ) )
82, 7mpan9 456 1  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666  (class class class)co 6040
This theorem is referenced by:  caovcld  6199  caovcl  6200  grprinvd  6245  seqcl2  11296  seqcaopr  11315  ercpbl  13729  imasmnd2  14687  imasgrp2  14888  gsumzaddlem  15481  imasrng  15680  divsrhm  16263  mplind  16517  plymullem  20088  gsumpropd2lem  24173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
  Copyright terms: Public domain W3C validator