Metamath Proof Explorer

Theorem pprodcnveq

Description: A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014)

Ref Expression
Assertion pprodcnveq 
|- pprod ( R , S ) = `' pprod ( `' R , `' S )

Proof

Step Hyp Ref Expression
1 dfpprod2 
 |-  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 
 |-  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 
 |-  `' 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 
 |-  `' ( ( `' ( 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 
 |-  `' ( `' ( 1st |` ( _V X. _V ) ) o. ( `' R o. ( 1st |` ( _V X. _V ) ) ) ) = ( `' ( `' R o. ( 1st |` ( _V X. _V ) ) ) o. ( 1st |` ( _V X. _V ) ) )
6 cnvco1 
 |-  `' ( `' R o. ( 1st |` ( _V X. _V ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. R )
7 6 coeq1i 
 |-  ( `' ( `' R o. ( 1st |` ( _V X. _V ) ) ) o. ( 1st |` ( _V X. _V ) ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. R ) o. ( 1st |` ( _V X. _V ) ) )
8 coass 
 |-  ( ( `' ( 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 
 |-  `' ( `' ( 1st |` ( _V X. _V ) ) o. ( `' R o. ( 1st |` ( _V X. _V ) ) ) ) = ( `' ( 1st |` ( _V X. _V ) ) o. ( R o. ( 1st |` ( _V X. _V ) ) ) )
10 cnvco1 
 |-  `' ( `' ( 2nd |` ( _V X. _V ) ) o. ( `' S o. ( 2nd |` ( _V X. _V ) ) ) ) = ( `' ( `' S o. ( 2nd |` ( _V X. _V ) ) ) o. ( 2nd |` ( _V X. _V ) ) )
11 cnvco1 
 |-  `' ( `' S o. ( 2nd |` ( _V X. _V ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. S )
12 11 coeq1i 
 |-  ( `' ( `' S o. ( 2nd |` ( _V X. _V ) ) ) o. ( 2nd |` ( _V X. _V ) ) ) = ( ( `' ( 2nd |` ( _V X. _V ) ) o. S ) o. ( 2nd |` ( _V X. _V ) ) )
13 coass 
 |-  ( ( `' ( 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 
 |-  `' ( `' ( 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 
 |-  ( `' ( `' ( 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 
 |-  `' 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 
 |-  pprod ( R , S ) = `' pprod ( `' R , `' S )