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Theorem mulpqnq 9315
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 9289 . . . . 5  |-  .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5865 . . . 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 5030 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5878 . . . 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 9283 . . . . 5  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
8 opex 4711 . . . . 5  |-  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  e.  _V
97, 8fnmpt2i 6850 . . . 4  |-  .pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 9299 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 9299 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 5030 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  <. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
1310, 11, 12syl2an 477 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
14 fvco2 5940 . . . 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 663 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( /Q  o.  .pQ  ) `  <. A ,  B >. )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) ) )
163, 6, 153eqtrd 2512 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  .Q  `  <. A ,  B >. )  =  ( /Q `  (  .pQ  `  <. A ,  B >. ) ) )
17 df-ov 6285 . 2  |-  ( A  .Q  B )  =  (  .Q  `  <. A ,  B >. )
18 df-ov 6285 . . 3  |-  ( A 
.pQ  B )  =  (  .pQ  `  <. A ,  B >. )
1918fveq2i 5867 . 2  |-  ( /Q
`  ( A  .pQ  B ) )  =  ( /Q `  (  .pQ  ` 
<. A ,  B >. ) )
2016, 17, 193eqtr4g 2533 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 369    = wceq 1379    e. wcel 1767   <.cop 4033    X. cxp 4997    |` cres 5001    o. ccom 5003    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   N.cnpi 9218    .N cmi 9220    .pQ cmpq 9223   Q.cnq 9226   /Qcerq 9228    .Q cmq 9230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-mpq 9283  df-nq 9286  df-mq 9289
This theorem is referenced by:  mulclnq  9321  mulcomnq  9327  mulerpq  9331  mulassnq  9333  distrnq  9335  mulidnq  9337  ltmnq  9346
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