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Theorem addpipq2 9220
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 5802 . . . . 5  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21oveq1d 6218 . . . 4  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  A
)  .N  ( 2nd `  y ) ) )
3 fveq2 5802 . . . . 5  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
43oveq2d 6219 . . . 4  |-  ( x  =  A  ->  (
( 1st `  y
)  .N  ( 2nd `  x ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  A ) ) )
52, 4oveq12d 6221 . . 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 6218 . . 3  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
75, 6opeq12d 4178 . 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 5802 . . . . 5  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
98oveq2d 6219 . . . 4  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )
10 fveq2 5802 . . . . 5  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
1110oveq1d 6218 . . . 4  |-  ( y  =  B  ->  (
( 1st `  y
)  .N  ( 2nd `  A ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
129, 11oveq12d 6221 . . 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 6219 . . 3  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
1412, 13opeq12d 4178 . 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 9192 . 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 4667 . 2  |-  <. (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  e.  _V
177, 14, 15, 16ovmpt2 6339 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 1370    e. wcel 1758   <.cop 3994    X. cxp 4949   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   N.cnpi 9126    +N cpli 9127    .N cmi 9128    +pQ cplpq 9130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-plpq 9192
This theorem is referenced by:  addpipq  9221  addcompq  9234  adderpqlem  9238  addassnq  9242  distrnq  9245  ltanq  9255
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