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Theorem mulpqnq 9267
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
Assertion
Ref Expression
mulpqnq  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )

Proof of Theorem mulpqnq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mq 9241 . . . . 5  |-  .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5804 . . . 4  |-  (  .Q 
`  <. A ,  B >. )  =  ( ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )
32a1i 11 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  .Q  `  <. A ,  B >. )  =  ( ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )
)
4 opelxpi 4972 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5817 . . . 4  |-  ( <. A ,  B >.  e.  ( Q.  X.  Q. )  ->  ( ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )  =  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. ) )
64, 5syl 17 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) `
 <. A ,  B >. )  =  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. ) )
7 df-mpq 9235 . . . . 5  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
8 opex 4652 . . . . 5  |-  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  e.  _V
97, 8fnmpt2i 6805 . . . 4  |-  .pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 9251 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 9251 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 4972 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  <. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
1310, 11, 12syl2an 475 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
14 fvco2 5878 . . . 4  |-  ( ( 
.pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )  /\  <. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )  -> 
( ( /Q  o.  .pQ  ) `  <. A ,  B >. )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) ) )
159, 13, 14sylancr 661 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) ) )
163, 6, 153eqtrd 2445 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  .Q  `  <. A ,  B >. )  =  ( /Q `  (  .pQ  `  <. A ,  B >. ) ) )
17 df-ov 6235 . 2  |-  ( A  .Q  B )  =  (  .Q  `  <. A ,  B >. )
18 df-ov 6235 . . 3  |-  ( A 
.pQ  B )  =  (  .pQ  `  <. A ,  B >. )
1918fveq2i 5806 . 2  |-  ( /Q
`  ( A  .pQ  B ) )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) )
2016, 17, 193eqtr4g 2466 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   <.cop 3975    X. cxp 4938    |` cres 4942    o. ccom 4944    Fn wfn 5518   ` cfv 5523  (class class class)co 6232   1stc1st 6734   2ndc2nd 6735   N.cnpi 9170    .N cmi 9172    .pQ cmpq 9175   Q.cnq 9178   /Qcerq 9180    .Q cmq 9182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-mpq 9235  df-nq 9238  df-mq 9241
This theorem is referenced by:  mulclnq  9273  mulcomnq  9279  mulerpq  9283  mulassnq  9285  distrnq  9287  mulidnq  9289  ltmnq  9298
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