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Theorem pprodcnveq 24494
Description: A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
pprodcnveq  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )

Proof of Theorem pprodcnveq
StepHypRef Expression
1 dfpprod2 24493 . 2  |- pprod ( R ,  S )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
2 dfpprod2 24493 . . . 4  |- pprod ( `' R ,  `' S
)  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
32cnveqi 4872 . . 3  |-  `'pprod ( `' R ,  `' S
)  =  `' ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
4 cnvin 5104 . . 3  |-  `' ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )  =  ( `' ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  `' ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
5 cnvco1 24188 . . . . 5  |-  `' ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  ( `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )
6 cnvco1 24188 . . . . . 6  |-  `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R
)
76coeq1i 4859 . . . . 5  |-  ( `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R )  o.  ( 1st  |`  ( _V  X.  _V ) ) )
8 coass 5207 . . . . 5  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )
95, 7, 83eqtri 2320 . . . 4  |-  `' ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )
10 cnvco1 24188 . . . . 5  |-  `' ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  ( `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )
11 cnvco1 24188 . . . . . 6  |-  `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S
)
1211coeq1i 4859 . . . . 5  |-  ( `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )
13 coass 5207 . . . . 5  |-  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
1410, 12, 133eqtri 2320 . . . 4  |-  `' ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
159, 14ineq12i 3381 . . 3  |-  ( `' ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  `' ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
163, 4, 153eqtri 2320 . 2  |-  `'pprod ( `' R ,  `' S
)  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( R  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( S  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
171, 16eqtr4i 2319 1  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
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  pprodcpprod 24445
This theorem is referenced by:  brpprod3b  24498
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-txp 24466  df-pprod 24467
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