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Theorem mulpipq2 9320
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
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 of Theorem mulpipq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5856 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21oveq1d 6296 . . 3  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  y ) ) )
3 fveq2 5856 . . . 4  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
43oveq1d 6296 . . 3  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
52, 4opeq12d 4210 . 2  |-  ( x  =  A  ->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  =  <. ( ( 1st `  A )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  y ) )
>. )
6 fveq2 5856 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
76oveq2d 6297 . . 3  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  B ) ) )
8 fveq2 5856 . . . 4  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
98oveq2d 6297 . . 3  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
107, 9opeq12d 4210 . 2  |-  ( 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 9290 . 2  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
12 opex 4701 . 2  |-  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  e.  _V
135, 10, 11, 12ovmpt2 6423 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   <.cop 4020    X. cxp 4987   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   N.cnpi 9225    .N cmi 9227    .pQ cmpq 9230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-mpq 9290
This theorem is referenced by:  mulpipq  9321  mulcompq  9333  mulerpqlem  9336  mulassnq  9340  distrnq  9342  ltmnq  9353
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