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Theorem addpqf 8747
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 6308 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
2 xp2nd 6309 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
3 mulclpi 8696 . . . . . 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 6308 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
6 xp2nd 6309 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
7 mulclpi 8696 . . . . . 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 8695 . . . . 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 8696 . . . . 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 4843 . . . 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 2708 . 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 8711 . . 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 6350 . 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 1717   A.wral 2642   <.cop 3753    X. cxp 4809   -->wf 5383   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   N.cnpi 8645    +N cpli 8646    .N cmi 8647    +pQ cplpq 8649
This theorem is referenced by:  addclnq  8748  addnqf  8751  addcompq  8753  adderpq  8759  distrnq  8764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-oadd 6657  df-omul 6658  df-ni 8675  df-pli 8676  df-mi 8677  df-plpq 8711
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