ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulpipq2 Unicode version

Theorem mulpipq2 7186
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
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 xp1st 6063 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
2 xp1st 6063 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
3 mulclpi 7143 . . . 4  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
41, 2, 3syl2an 287 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
5 xp2nd 6064 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
6 xp2nd 6064 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
7 mulclpi 7143 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
85, 6, 7syl2an 287 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
9 opexg 4150 . . 3  |-  ( ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N.  /\  ( ( 2nd `  A )  .N  ( 2nd `  B
) )  e.  N. )  ->  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  e.  _V )
104, 8, 9syl2anc 408 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  e.  _V )
11 fveq2 5421 . . . . 5  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
1211oveq1d 5789 . . . 4  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  y ) ) )
13 fveq2 5421 . . . . 5  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
1413oveq1d 5789 . . . 4  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
1512, 14opeq12d 3713 . . 3  |-  ( x  =  A  ->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  =  <. ( ( 1st `  A )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  y ) )
>. )
16 fveq2 5421 . . . . 5  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
1716oveq2d 5790 . . . 4  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  B ) ) )
18 fveq2 5421 . . . . 5  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
1918oveq2d 5790 . . . 4  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
2017, 19opeq12d 3713 . . 3  |-  ( y  =  B  ->  <. (
( 1st `  A
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) >.  =  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>. )
21 df-mpq 7160 . . 3  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
2215, 20, 21ovmpog 5905 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  e.  _V )  -> 
( A  .pQ  B
)  =  <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
2310, 22mpd3an3 1316 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 set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686   <.cop 3530    X. cxp 4537   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   N.cnpi 7087    .N cmi 7089    .pQ cmpq 7092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-ni 7119  df-mi 7121  df-mpq 7160
This theorem is referenced by:  mulpipq  7187
  Copyright terms: Public domain W3C validator