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Theorem mulpqnq 8744
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 8718 . . . . 5  |-  .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5662 . . . 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 4843 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5678 . . . 4  |-  ( <. A ,  B >.  e.  ( Q.  X.  Q. )  ->  ( ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )  =  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. ) )
64, 5syl 16 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) `
 <. A ,  B >. )  =  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. ) )
7 df-mpq 8712 . . . . 5  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
8 opex 4361 . . . . 5  |-  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  e.  _V
97, 8fnmpt2i 6352 . . . 4  |-  .pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 8728 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 8728 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 4843 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  <. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
1310, 11, 12syl2an 464 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
14 fvco2 5730 . . . 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 645 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) ) )
163, 6, 153eqtrd 2416 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  .Q  `  <. A ,  B >. )  =  ( /Q `  (  .pQ  `  <. A ,  B >. ) ) )
17 df-ov 6016 . 2  |-  ( A  .Q  B )  =  (  .Q  `  <. A ,  B >. )
18 df-ov 6016 . . 3  |-  ( A 
.pQ  B )  =  (  .pQ  `  <. A ,  B >. )
1918fveq2i 5664 . 2  |-  ( /Q
`  ( A  .pQ  B ) )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) )
2016, 17, 193eqtr4g 2437 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3753    X. cxp 4809    |` cres 4813    o. ccom 4815    Fn wfn 5382   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   N.cnpi 8645    .N cmi 8647    .pQ cmpq 8650   Q.cnq 8653   /Qcerq 8655    .Q cmq 8657
This theorem is referenced by:  mulclnq  8750  mulcomnq  8756  mulerpq  8760  mulassnq  8762  distrnq  8764  mulidnq  8766  ltmnq  8775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-mpq 8712  df-nq 8715  df-mq 8718
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