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

Theorem mulpipq2 6623
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 5823 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
2 xp1st 5823 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
3 mulclpi 6580 . . . 4  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
41, 2, 3syl2an 283 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
5 xp2nd 5824 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
6 xp2nd 5824 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
7 mulclpi 6580 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
85, 6, 7syl2an 283 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
9 opexg 3991 . . 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 403 . 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 5209 . . . . 5  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
1211oveq1d 5558 . . . 4  |-  ( x  =  A  ->  (
( 1st `  x
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  y ) ) )
13 fveq2 5209 . . . . 5  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
1413oveq1d 5558 . . . 4  |-  ( x  =  A  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  y ) ) )
1512, 14opeq12d 3586 . . 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 5209 . . . . 5  |-  ( y  =  B  ->  ( 1st `  y )  =  ( 1st `  B
) )
1716oveq2d 5559 . . . 4  |-  ( y  =  B  ->  (
( 1st `  A
)  .N  ( 1st `  y ) )  =  ( ( 1st `  A
)  .N  ( 1st `  B ) ) )
18 fveq2 5209 . . . . 5  |-  ( y  =  B  ->  ( 2nd `  y )  =  ( 2nd `  B
) )
1918oveq2d 5559 . . . 4  |-  ( y  =  B  ->  (
( 2nd `  A
)  .N  ( 2nd `  y ) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )
2017, 19opeq12d 3586 . . 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 6597 . . 3  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
2215, 20, 21ovmpt2g 5666 . 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 1270 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 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   <.cop 3409    X. cxp 4369   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   N.cnpi 6524    .N cmi 6526    .pQ cmpq 6529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070  df-ni 6556  df-mi 6558  df-mpq 6597
This theorem is referenced by:  mulpipq  6624
  Copyright terms: Public domain W3C validator