Description: Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dfpprod2 | |- pprod ( A , B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. ( A o. ( 1st |` ( _V X. _V ) ) ) ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. ( B o. ( 2nd |` ( _V X. _V ) ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pprod | |- pprod ( A , B ) = ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) |
|
| 2 | df-txp | |- ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. ( A o. ( 1st |` ( _V X. _V ) ) ) ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. ( B o. ( 2nd |` ( _V X. _V ) ) ) ) ) |
|
| 3 | 1 2 | eqtri | |- pprod ( A , B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. ( A o. ( 1st |` ( _V X. _V ) ) ) ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. ( B o. ( 2nd |` ( _V X. _V ) ) ) ) ) |