MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addpipq2 Structured version   Unicode version

Theorem addpipq2 9326
Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addpipq2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  +pQ  B )  = 
<. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )

Proof of Theorem addpipq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . 5  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21oveq1d 6310 . . . 4  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  A
)  .N  ( 2nd `  y ) ) )
3 fveq2 5872 . . . . 5  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
43oveq2d 6311 . . . 4  |-  ( x  =  A  ->  (
( 1st `  y
)  .N  ( 2nd `  x ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  A ) ) )
52, 4oveq12d 6313 . . 3  |-  ( x  =  A  ->  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  A ) ) ) )
63oveq1d 6310 . . 3  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
75, 6opeq12d 4227 . 2  |-  ( x  =  A  ->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  =  <. ( ( ( 1st `  A
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  y ) )
>. )
8 fveq2 5872 . . . . 5  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
98oveq2d 6311 . . . 4  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )
10 fveq2 5872 . . . . 5  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
1110oveq1d 6310 . . . 4  |-  ( y  =  B  ->  (
( 1st `  y
)  .N  ( 2nd `  A ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
129, 11oveq12d 6313 . . 3  |-  ( y  =  B  ->  (
( ( 1st `  A
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
138oveq2d 6311 . . 3  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
1412, 13opeq12d 4227 . 2  |-  ( y  =  B  ->  <. (
( ( 1st `  A
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  y ) )
>.  =  <. ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )
15 df-plpq 9298 . 2  |-  +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 ) )
>. )
16 opex 4717 . 2  |-  <. (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  e.  _V
177, 14, 15, 16ovmpt2 6433 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  +pQ  B )  = 
<. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4039    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   N.cnpi 9234    +N cpli 9235    .N cmi 9236    +pQ cplpq 9238
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
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-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-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-plpq 9298
This theorem is referenced by:  addpipq  9327  addcompq  9340  adderpqlem  9344  addassnq  9348  distrnq  9351  ltanq  9361
  Copyright terms: Public domain W3C validator