Theorem pprodcnveq 31990

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

Step	Hyp	Ref	Expression
1		dfpprod2 31989	. 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 31989	. . . 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 ) ) ) ) )
3	2	cnveqi 5297	. . 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 5540	. . 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 31649	. . . . 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 31649	. . . . . 6 |- `' ( `' R o. ( 1st |` ( _V X. _V ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. R )
7	6	coeq1i 5281	. . . . 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 5654	. . . . 5 |- ( ( `' ( 1st |` ( _V X. _V ) ) o. R ) o. ( 1st |` ( _V X. _V ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. ( R o. ( 1st |` ( _V X. _V ) ) ) )
9	5, 7, 8	3eqtri 2648	. . . 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 31649	. . . . 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 31649	. . . . . 6 |- `' ( `' S o. ( 2nd |` ( _V X. _V ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. S )
12	11	coeq1i 5281	. . . . 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 5654	. . . . 5 |- ( ( `' ( 2nd |` ( _V X. _V ) ) o. S ) o. ( 2nd |` ( _V X. _V ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. ( S o. ( 2nd |` ( _V X. _V ) ) ) )
14	10, 12, 13	3eqtri 2648	. . . 4 |- `' ( `' ( 2nd |` ( _V X. _V ) ) o. ( `' S o. ( 2nd |` ( _V X. _V ) ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. ( S o. ( 2nd |` ( _V X. _V ) ) ) )
15	9, 14	ineq12i 3812	. . 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 ) ) ) ) )
16	3, 4, 15	3eqtri 2648	. 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 ) ) ) ) )
17	1, 16	eqtr4i 2647	1 |- pprod ( R , S ) = `'pprod ( `' R , `' S )

Syntax hints:   = wceq 1483   _Vcvv 3200   i^i cin 3573   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   1stc1st 7166   2ndc2nd 7167  pprodcpprod 31938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-txp 31961  df-pprod 31962

This theorem is referenced by:  brpprod3b  31994