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Theorem addpipq2 8576
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 5541 . . . . 5  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21oveq1d 5889 . . . 4  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  A
)  .N  ( 2nd `  y ) ) )
3 fveq2 5541 . . . . 5  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
43oveq2d 5890 . . . 4  |-  ( x  =  A  ->  (
( 1st `  y
)  .N  ( 2nd `  x ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  A ) ) )
52, 4oveq12d 5892 . . 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 5889 . . 3  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
75, 6opeq12d 3820 . 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 5541 . . . . 5  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
98oveq2d 5890 . . . 4  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )
10 fveq2 5541 . . . . 5  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
1110oveq1d 5889 . . . 4  |-  ( y  =  B  ->  (
( 1st `  y
)  .N  ( 2nd `  A ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
129, 11oveq12d 5892 . . 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 5890 . . 3  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
1412, 13opeq12d 3820 . 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 8548 . 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 4253 . 2  |-  <. (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  e.  _V
177, 14, 15, 16ovmpt2 5999 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 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:  addpipq  8577  addcompq  8590  adderpqlem  8594  addassnq  8598  distrnq  8601  ltanq  8611
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-plpq 8548
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