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Theorem mulpipq2 9318
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpipq2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)

Proof of Theorem mulpipq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5866 . . . 4  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
21oveq1d 6300 . . 3  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  y ) ) )
3 fveq2 5866 . . . 4  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
43oveq1d 6300 . . 3  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
52, 4opeq12d 4221 . 2  |-  ( x  =  A  ->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  =  <. ( ( 1st `  A )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  y ) )
>. )
6 fveq2 5866 . . . 4  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
76oveq2d 6301 . . 3  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  B ) ) )
8 fveq2 5866 . . . 4  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
98oveq2d 6301 . . 3  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
107, 9opeq12d 4221 . 2  |-  ( y  =  B  ->  <. (
( 1st `  A
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) >.  =  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )
11 df-mpq 9288 . 2  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
12 opex 4711 . 2  |-  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  e.  _V
135, 10, 11, 12ovmpt2 6423 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033    X. cxp 4997   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   N.cnpi 9223    .N cmi 9225    .pQ cmpq 9228
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 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-mpq 9288
This theorem is referenced by:  mulpipq  9319  mulcompq  9331  mulerpqlem  9334  mulassnq  9338  distrnq  9340  ltmnq  9351
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