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Theorem addpqf 8810
Description: Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addpqf  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )

Proof of Theorem addpqf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6367 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
2 xp2nd 6368 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
3 mulclpi 8759 . . . . . 6  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
41, 2, 3syl2an 464 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
5 xp1st 6367 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
6 xp2nd 6368 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
7 mulclpi 8759 . . . . . 6  |-  ( ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
85, 6, 7syl2anr 465 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
9 addclpi 8758 . . . . 5  |-  ( ( ( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N.  /\  ( ( 1st `  y )  .N  ( 2nd `  x
) )  e.  N. )  ->  ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) )  e.  N. )
104, 8, 9syl2anc 643 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )  e.  N. )
11 mulclpi 8759 . . . . 5  |-  ( ( ( 2nd `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
126, 2, 11syl2an 464 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
13 opelxpi 4901 . . . 4  |-  ( ( ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )  e.  N.  /\  ( ( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )  ->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  ( N.  X.  N. ) )
1410, 12, 13syl2anc 643 . . 3  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  ( N.  X.  N. ) )
1514rgen2a 2764 . 2  |-  A. x  e.  ( N.  X.  N. ) A. y  e.  ( N.  X.  N. ) <. ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y
) ) >.  e.  ( N.  X.  N. )
16 df-plpq 8774 . . 3  |-  +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
1716fmpt2 6409 . 2  |-  ( A. x  e.  ( N.  X.  N. ) A. y  e.  ( N.  X.  N. ) <. ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.  e.  ( N.  X.  N. )  <->  +pQ  : ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N. 
X.  N. ) )
1815, 17mpbi 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 2697   <.cop 3809    X. cxp 4867   -->wf 5441   ` cfv 5445  (class class class)co 6072   1stc1st 6338   2ndc2nd 6339   N.cnpi 8708    +N cpli 8709    .N cmi 8710    +pQ cplpq 8712
This theorem is referenced by:  addclnq  8811  addnqf  8814  addcompq  8816  adderpq  8822  distrnq  8827
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-oadd 6719  df-omul 6720  df-ni 8738  df-pli 8739  df-mi 8740  df-plpq 8774
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