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Theorem mulpqf 9327
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 6815 . . . . 5  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
2 xp1st 6815 . . . . 5  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
3 mulclpi 9274 . . . . 5  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 1st `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 1st `  y ) )  e. 
N. )
41, 2, 3syl2an 477 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 1st `  x
)  .N  ( 1st `  y ) )  e. 
N. )
5 xp2nd 6816 . . . . 5  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
6 xp2nd 6816 . . . . 5  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
7 mulclpi 9274 . . . . 5  |-  ( ( ( 2nd `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
85, 6, 7syl2an 477 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
9 opelxpi 5021 . . . 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 661 . . 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 2870 . 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 9290 . . 3  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
1312fmpt2 6852 . 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 208 1  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1804   A.wral 2793   <.cop 4020    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   N.cnpi 9225    .N cmi 9227    .pQ cmpq 9230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-oadd 7136  df-omul 7137  df-ni 9253  df-mi 9255  df-mpq 9290
This theorem is referenced by:  mulclnq  9328  mulnqf  9330  mulcompq  9333  mulerpq  9338  distrnq  9342
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