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

Theorem pprodcnveq 30189
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 30188 . 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 30188 . . . 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 5117 . . 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 5350 . . 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 29854 . . . . 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 29854 . . . . . 6  |-  `' ( `' R  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  R
)
76coeq1i 5102 . . . . 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 5461 . . . . 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 2433 . . . 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 29854 . . . . 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 29854 . . . . . 6  |-  `' ( `' S  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  S
)
1211coeq1i 5102 . . . . 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 5461 . . . . 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 2433 . . . 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 3636 . . 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 2433 . 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 2432 1  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1403   _Vcvv 3056    i^i cin 3410    X. cxp 4938   `'ccnv 4939    |` cres 4942    o. ccom 4944   1stc1st 6734   2ndc2nd 6735  pprodcpprod 30136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-txp 30159  df-pprod 30160
This theorem is referenced by:  brpprod3b  30193
  Copyright terms: Public domain W3C validator