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Theorem addpqnq 9328
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 9304 . . . . 5  |-  +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5873 . . . 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 5037 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5886 . . . 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 9298 . . . . 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 4717 . . . . 5  |-  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  _V
97, 8fnmpt2i 6864 . . . 4  |-  +pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 9315 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 9315 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 5037 . . . . 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 5949 . . . 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 6298 . 2  |-  ( A  +Q  B )  =  (  +Q  `  <. A ,  B >. )
18 df-ov 6298 . . 3  |-  ( A 
+pQ  B )  =  (  +pQ  `  <. A ,  B >. )
1918fveq2i 5875 . 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 4039    X. cxp 5003    |` cres 5007    o. ccom 5009    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   N.cnpi 9234    +N cpli 9235    .N cmi 9236    +pQ cplpq 9238   Q.cnq 9242   /Qcerq 9244    +Q cplq 9245
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-plpq 9298  df-nq 9302  df-plq 9304
This theorem is referenced by:  addclnq  9335  addcomnq  9341  adderpq  9346  addassnq  9348  distrnq  9351  ltanq  9361  1lt2nq  9363  prlem934  9423
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