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Theorem addpqnq 8607
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 8583 . . . . 5  |-  +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q. 
X.  Q. ) )
21fveq1i 5564 . . . 4  |-  (  +Q 
`  <. A ,  B >. )  =  ( ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )
32a1i 10 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  +Q  `  <. A ,  B >. )  =  ( ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )
)
4 opelxpi 4758 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( Q.  X.  Q. ) )
5 fvres 5580 . . . 4  |-  ( <. A ,  B >.  e.  ( Q.  X.  Q. )  ->  ( ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) `  <. A ,  B >. )  =  ( ( /Q  o.  +pQ  ) `  <. A ,  B >. ) )
64, 5syl 15 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) `
 <. A ,  B >. )  =  ( ( /Q  o.  +pQ  ) `  <. A ,  B >. ) )
7 df-plpq 8577 . . . . 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 4274 . . . . 5  |-  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  _V
97, 8fnmpt2i 6235 . . . 4  |-  +pQ  Fn  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
10 elpqn 8594 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
11 elpqn 8594 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
12 opelxpi 4758 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  <. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
1310, 11, 12syl2an 463 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. A ,  B >.  e.  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
14 fvco2 5632 . . . 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 644 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( ( /Q  o.  +pQ  ) `  <. A ,  B >. )  =  ( /Q `  (  +pQ  ` 
<. A ,  B >. ) ) )
163, 6, 153eqtrd 2352 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  (  +Q  `  <. A ,  B >. )  =  ( /Q `  (  +pQ  `  <. A ,  B >. ) ) )
17 df-ov 5903 . 2  |-  ( A  +Q  B )  =  (  +Q  `  <. A ,  B >. )
18 df-ov 5903 . . 3  |-  ( A 
+pQ  B )  =  (  +pQ  `  <. A ,  B >. )
1918fveq2i 5566 . 2  |-  ( /Q
`  ( A  +pQ  B ) )  =  ( /Q `  (  +pQ  ` 
<. A ,  B >. ) )
2016, 17, 193eqtr4g 2373 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 358    = wceq 1633    e. wcel 1701   <.cop 3677    X. cxp 4724    |` cres 4728    o. ccom 4730    Fn wfn 5287   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   N.cnpi 8511    +N cpli 8512    .N cmi 8513    +pQ cplpq 8515   Q.cnq 8519   /Qcerq 8521    +Q cplq 8522
This theorem is referenced by:  addclnq  8614  addcomnq  8620  adderpq  8625  addassnq  8627  distrnq  8630  ltanq  8640  1lt2nq  8642  prlem934  8702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-plpq 8577  df-nq 8581  df-plq 8583
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