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Theorem addpipq 9332
Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addpipq  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  +pQ  <. C ,  D >. )  =  <. ( ( A  .N  D
)  +N  ( C  .N  B ) ) ,  ( B  .N  D ) >. )

Proof of Theorem addpipq
StepHypRef Expression
1 opelxpi 5040 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 opelxpi 5040 . . 3  |-  ( ( C  e.  N.  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( N.  X.  N. ) )
3 addpipq2 9331 . . 3  |-  ( (
<. A ,  B >.  e.  ( N.  X.  N. )  /\  <. C ,  D >.  e.  ( N.  X.  N. ) )  ->  ( <. A ,  B >.  +pQ 
<. C ,  D >. )  =  <. ( ( ( 1st `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) )  +N  (
( 1st `  <. C ,  D >. )  .N  ( 2nd `  <. A ,  B >. )
) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >. )
41, 2, 3syl2an 477 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  +pQ  <. C ,  D >. )  =  <. ( ( ( 1st `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
)  +N  ( ( 1st `  <. C ,  D >. )  .N  ( 2nd `  <. A ,  B >. ) ) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >. )
5 op1stg 6811 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 1st `  <. A ,  B >. )  =  A )
6 op2ndg 6812 . . . . 5  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 2nd `  <. C ,  D >. )  =  D )
75, 6oveqan12d 6315 . . . 4  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 1st `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) )  =  ( A  .N  D ) )
8 op1stg 6811 . . . . 5  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 1st `  <. C ,  D >. )  =  C )
9 op2ndg 6812 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
108, 9oveqan12rd 6316 . . . 4  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 1st `  <. C ,  D >. )  .N  ( 2nd `  <. A ,  B >. ) )  =  ( C  .N  B ) )
117, 10oveq12d 6314 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( (
( 1st `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
)  +N  ( ( 1st `  <. C ,  D >. )  .N  ( 2nd `  <. A ,  B >. ) ) )  =  ( ( A  .N  D )  +N  ( C  .N  B ) ) )
129, 6oveqan12d 6315 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. ) )  =  ( B  .N  D ) )
1311, 12opeq12d 4227 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  <. ( ( ( 1st `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
)  +N  ( ( 1st `  <. C ,  D >. )  .N  ( 2nd `  <. A ,  B >. ) ) ) ,  ( ( 2nd `  <. A ,  B >. )  .N  ( 2nd `  <. C ,  D >. )
) >.  =  <. (
( A  .N  D
)  +N  ( C  .N  B ) ) ,  ( B  .N  D ) >. )
144, 13eqtrd 2498 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
)  ->  ( <. A ,  B >.  +pQ  <. C ,  D >. )  =  <. ( ( A  .N  D
)  +N  ( C  .N  B ) ) ,  ( B  .N  D ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   <.cop 4038    X. cxp 5006   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   N.cnpi 9239    +N cpli 9240    .N cmi 9241    +pQ cplpq 9243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-plpq 9303
This theorem is referenced by:  addassnq  9353  distrnq  9356  1lt2nq  9368  ltexnq  9370  prlem934  9428
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