Metamath Proof Explorer


Theorem mulpipq2

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013) (New usage is discouraged.)

Ref Expression
Assertion mulpipq2
|- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )

Proof

Step Hyp Ref Expression
1 fveq2
 |-  ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
2 1 oveq1d
 |-  ( x = A -> ( ( 1st ` x ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` y ) ) )
3 fveq2
 |-  ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
4 3 oveq1d
 |-  ( x = A -> ( ( 2nd ` x ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` y ) ) )
5 2 4 opeq12d
 |-  ( x = A -> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. = <. ( ( 1st ` A ) .N ( 1st ` y ) ) , ( ( 2nd ` A ) .N ( 2nd ` y ) ) >. )
6 fveq2
 |-  ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
7 6 oveq2d
 |-  ( y = B -> ( ( 1st ` A ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` B ) ) )
8 fveq2
 |-  ( y = B -> ( 2nd ` y ) = ( 2nd ` B ) )
9 8 oveq2d
 |-  ( y = B -> ( ( 2nd ` A ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
10 7 9 opeq12d
 |-  ( y = B -> <. ( ( 1st ` A ) .N ( 1st ` y ) ) , ( ( 2nd ` A ) .N ( 2nd ` y ) ) >. = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
11 df-mpq
 |-  .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
12 opex
 |-  <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. e. _V
13 5 10 11 12 ovmpo
 |-  ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )