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Theorem addpipq 8577
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 4737 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 opelxpi 4737 . . 3  |-  ( ( C  e.  N.  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( N.  X.  N. ) )
3 addpipq2 8576 . . 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 463 . 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 6148 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 1st `  <. A ,  B >. )  =  A )
6 op2ndg 6149 . . . . 5  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 2nd `  <. C ,  D >. )  =  D )
75, 6oveqan12d 5893 . . . 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 6148 . . . . 5  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 1st `  <. C ,  D >. )  =  C )
9 op2ndg 6149 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
108, 9oveqan12rd 5894 . . . 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 5892 . . 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 5893 . . 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 3820 . 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 2328 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   N.cnpi 8482    +N cpli 8483    .N cmi 8484    +pQ cplpq 8486
This theorem is referenced by:  addassnq  8598  distrnq  8601  1lt2nq  8613  ltexnq  8615  prlem934  8673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-plpq 8548
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