MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulpqnq Structured version   Unicode version

Theorem mulpqnq 8810
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 8784 . . . . 5  |-  .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5721 . . . 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 4902 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5737 . . . 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 8778 . . . . 5  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
8 opex 4419 . . . . 5  |-  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  e.  _V
97, 8fnmpt2i 6412 . . . 4  |-  .pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 8794 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 8794 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 4902 . . . . 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 5790 . . . 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 2471 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  .Q  `  <. A ,  B >. )  =  ( /Q `  (  .pQ  `  <. A ,  B >. ) ) )
17 df-ov 6076 . 2  |-  ( A  .Q  B )  =  (  .Q  `  <. A ,  B >. )
18 df-ov 6076 . . 3  |-  ( A 
.pQ  B )  =  (  .pQ  `  <. A ,  B >. )
1918fveq2i 5723 . 2  |-  ( /Q
`  ( A  .pQ  B ) )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) )
2016, 17, 193eqtr4g 2492 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 1652    e. wcel 1725   <.cop 3809    X. cxp 4868    |` cres 4872    o. ccom 4874    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   N.cnpi 8711    .N cmi 8713    .pQ cmpq 8716   Q.cnq 8719   /Qcerq 8721    .Q cmq 8723
This theorem is referenced by:  mulclnq  8816  mulcomnq  8822  mulerpq  8826  mulassnq  8828  distrnq  8830  mulidnq  8832  ltmnq  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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-mpq 8778  df-nq 8781  df-mq 8784
  Copyright terms: Public domain W3C validator