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Theorem mulpipq 8817
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpipq  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )

Proof of Theorem mulpipq
StepHypRef Expression
1 opelxpi 4910 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 opelxpi 4910 . . 3  |-  ( ( C  e.  N.  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( N.  X.  N. ) )
3 mulpipq2 8816 . . 3  |-  ( (
<. A ,  B >.  e.  ( N.  X.  N. )  /\  <. C ,  D >.  e.  ( N.  X.  N. ) )  ->  ( <. A ,  B >.  .pQ 
<. C ,  D >. )  =  <. ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. ) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >. )
41, 2, 3syl2an 464 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. )
) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) ) >. )
5 op1stg 6359 . . . 4  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 1st `  <. A ,  B >. )  =  A )
6 op1stg 6359 . . . 4  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 1st `  <. C ,  D >. )  =  C )
75, 6oveqan12d 6100 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. ) )  =  ( A  .N  C ) )
8 op2ndg 6360 . . . 4  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
9 op2ndg 6360 . . . 4  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 2nd `  <. C ,  D >. )  =  D )
108, 9oveqan12d 6100 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) )  =  ( B  .N  D ) )
117, 10opeq12d 3992 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  <. ( ( 1st `  <. A ,  B >. )  .N  ( 1st `  <. C ,  D >. ) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >.  =  <. ( A  .N  C ) ,  ( B  .N  D
) >. )
124, 11eqtrd 2468 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817    X. cxp 4876   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   N.cnpi 8719    .N cmi 8721    .pQ cmpq 8724
This theorem is referenced by:  mulassnq  8836  distrnq  8838  mulidnq  8840  recmulnq  8841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-mpq 8786
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