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Theorem addpqnq 8815
Description: Addition 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
addpqnq  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )

Proof of Theorem addpqnq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-plq 8791 . . . . 5  |-  +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5729 . . . 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 4910 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5745 . . . 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-plpq 8785 . . . . 5  |-  +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
8 opex 4427 . . . . 5  |-  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  _V
97, 8fnmpt2i 6420 . . . 4  |-  +pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 8802 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 8802 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 4910 . . . . 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 5798 . . . 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 2472 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  +Q  `  <. A ,  B >. )  =  ( /Q `  (  +pQ  `  <. A ,  B >. ) ) )
17 df-ov 6084 . 2  |-  ( A  +Q  B )  =  (  +Q  `  <. A ,  B >. )
18 df-ov 6084 . . 3  |-  ( A 
+pQ  B )  =  (  +pQ  `  <. A ,  B >. )
1918fveq2i 5731 . 2  |-  ( /Q
`  ( A  +pQ  B ) )  =  ( /Q `  (  +pQ  ` 
<. A ,  B >. ) )
2016, 17, 193eqtr4g 2493 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 3817    X. cxp 4876    |` cres 4880    o. ccom 4882    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   N.cnpi 8719    +N cpli 8720    .N cmi 8721    +pQ cplpq 8723   Q.cnq 8727   /Qcerq 8729    +Q cplq 8730
This theorem is referenced by:  addclnq  8822  addcomnq  8828  adderpq  8833  addassnq  8835  distrnq  8838  ltanq  8848  1lt2nq  8850  prlem934  8910
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-csb 3252  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-iun 4095  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-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-plpq 8785  df-nq 8789  df-plq 8791
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