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Theorem pprodcnveq 25447
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 25446 . 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 25446 . . . 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 4987 . . 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 5219 . . 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 25141 . . . . 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 25141 . . . . . 6  |-  `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R
)
76coeq1i 4972 . . . . 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 5328 . . . . 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 2411 . . . 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 25141 . . . . 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 25141 . . . . . 6  |-  `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S
)
1211coeq1i 4972 . . . . 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 5328 . . . . 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 2411 . . . 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 3483 . . 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 2411 . 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 2410 1  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2899    i^i cin 3262    X. cxp 4816   `'ccnv 4817    |` cres 4820    o. ccom 4822   1stc1st 6286   2ndc2nd 6287  pprodcpprod 25398
This theorem is referenced by:  brpprod3b  25451
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-txp 25419  df-pprod 25420
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