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Theorem pprodcnveq 28078
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 28077 . 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 28077 . . . 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 5125 . . 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 5355 . . 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 27734 . . . . 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 27734 . . . . . 6  |-  `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R
)
76coeq1i 5110 . . . . 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 5467 . . . . 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 2487 . . . 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 27734 . . . . 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 27734 . . . . . 6  |-  `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S
)
1211coeq1i 5110 . . . . 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 5467 . . . . 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 2487 . . . 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 3661 . . 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 2487 . 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 2486 1  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3078    i^i cin 3438    X. cxp 4949   `'ccnv 4950    |` cres 4953    o. ccom 4955   1stc1st 6688   2ndc2nd 6689  pprodcpprod 28025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-txp 28048  df-pprod 28049
This theorem is referenced by:  brpprod3b  28082
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