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Theorem addpqf 9219
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 6711 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
2 xp2nd 6712 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
3 mulclpi 9168 . . . . . 6  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
41, 2, 3syl2an 477 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
5 xp1st 6711 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
6 xp2nd 6712 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
7 mulclpi 9168 . . . . . 6  |-  ( ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
85, 6, 7syl2anr 478 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
9 addclpi 9167 . . . . 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 661 . . . 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 9168 . . . . 5  |-  ( ( ( 2nd `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
126, 2, 11syl2an 477 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
13 opelxpi 4974 . . . 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 661 . . 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 2894 . 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 9183 . . 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 6746 . 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 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 1758   A.wral 2796   <.cop 3986    X. cxp 4941   -->wf 5517   ` cfv 5521  (class class class)co 6195   1stc1st 6680   2ndc2nd 6681   N.cnpi 9117    +N cpli 9118    .N cmi 9119    +pQ cplpq 9121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-oadd 7029  df-omul 7030  df-ni 9147  df-pli 9148  df-mi 9149  df-plpq 9183
This theorem is referenced by:  addclnq  9220  addnqf  9223  addcompq  9225  adderpq  9231  distrnq  9236
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