MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulpqf Structured version   Unicode version

Theorem mulpqf 8823
Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpqf  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )

Proof of Theorem mulpqf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6376 . . . . 5  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
2 xp1st 6376 . . . . 5  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
3 mulclpi 8770 . . . . 5  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 1st `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 1st `  y ) )  e. 
N. )
41, 2, 3syl2an 464 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 1st `  x
)  .N  ( 1st `  y ) )  e. 
N. )
5 xp2nd 6377 . . . . 5  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
6 xp2nd 6377 . . . . 5  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
7 mulclpi 8770 . . . . 5  |-  ( ( ( 2nd `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
85, 6, 7syl2an 464 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
9 opelxpi 4910 . . . 4  |-  ( ( ( ( 1st `  x
)  .N  ( 1st `  y ) )  e. 
N.  /\  ( ( 2nd `  x )  .N  ( 2nd `  y
) )  e.  N. )  ->  <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  ( N.  X.  N. ) )
104, 8, 9syl2anc 643 . . 3  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  e.  ( N.  X.  N. ) )
1110rgen2a 2772 . 2  |-  A. x  e.  ( N.  X.  N. ) A. y  e.  ( N.  X.  N. ) <. ( ( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.  e.  ( N.  X.  N. )
12 df-mpq 8786 . . 3  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
1312fmpt2 6418 . 2  |-  ( A. x  e.  ( N.  X.  N. ) A. y  e.  ( N.  X.  N. ) <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  ( N.  X.  N. )  <->  .pQ  : ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N. 
X.  N. ) )
1411, 13mpbi 200 1  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725   A.wral 2705   <.cop 3817    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   N.cnpi 8719    .N cmi 8721    .pQ cmpq 8724
This theorem is referenced by:  mulclnq  8824  mulnqf  8826  mulcompq  8829  mulerpq  8834  distrnq  8838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-oadd 6728  df-omul 6729  df-ni 8749  df-mi 8751  df-mpq 8786
  Copyright terms: Public domain W3C validator