Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfpprod2 Unicode version

Theorem dfpprod2 24493
Description: Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
dfpprod2  |- pprod ( A ,  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )

Proof of Theorem dfpprod2
StepHypRef Expression
1 df-pprod 24467 . 2  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
2 df-txp 24466 . 2  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
31, 2eqtri 2316 1  |- pprod ( A ,  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    i^i cin 3164    X. cxp 4703   `'ccnv 4704    |` cres 4707    o. ccom 4709   1stc1st 6136   2ndc2nd 6137    (x) ctxp 24444  pprodcpprod 24445
This theorem is referenced by:  pprodcnveq  24494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-cleq 2289  df-txp 24466  df-pprod 24467
  Copyright terms: Public domain W3C validator