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Theorem pprodcnveq 25763
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 25762 . 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 25762 . . . 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 5082 . . 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 5314 . . 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 25418 . . . . 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 25418 . . . . . 6  |-  `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R
)
76coeq1i 5067 . . . . 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 5423 . . . . 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 2467 . . . 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 25418 . . . . 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 25418 . . . . . 6  |-  `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S
)
1211coeq1i 5067 . . . . 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 5423 . . . . 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 2467 . . . 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 3529 . . 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 2467 . 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 2466 1  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
Colors of variables: wff set class
Syntax hints:    = wceq 1654   _Vcvv 2965    i^i cin 3308    X. cxp 4911   `'ccnv 4912    |` cres 4915    o. ccom 4917   1stc1st 6383   2ndc2nd 6384  pprodcpprod 25710
This theorem is referenced by:  brpprod3b  25767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-txp 25733  df-pprod 25734
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