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Theorem addpipq 9362
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 4866 . . 3  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 opelxpi 4866 . . 3  |-  ( ( C  e.  N.  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( N.  X.  N. ) )
3 addpipq2 9361 . . 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 480 . 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 6805 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 1st `  <. A ,  B >. )  =  A )
6 op2ndg 6806 . . . . 5  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 2nd `  <. C ,  D >. )  =  D )
75, 6oveqan12d 6309 . . . 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 6805 . . . . 5  |-  ( ( C  e.  N.  /\  D  e.  N. )  ->  ( 1st `  <. C ,  D >. )  =  C )
9 op2ndg 6806 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( 2nd `  <. A ,  B >. )  =  B )
108, 9oveqan12rd 6310 . . . 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 6308 . . 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 6309 . . 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 4174 . 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 2485 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 371    = wceq 1444    e. wcel 1887   <.cop 3974    X. cxp 4832   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   N.cnpi 9269    +N cpli 9270    .N cmi 9271    +pQ cplpq 9273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-plpq 9333
This theorem is referenced by:  addassnq  9383  distrnq  9386  1lt2nq  9398  ltexnq  9400  prlem934  9458
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