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Theorem addnqf 8856
Description: Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
addnqf  |-  +Q  :
( Q.  X.  Q. )
--> Q.

Proof of Theorem addnqf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nqerf 8838 . . . 4  |-  /Q :
( N.  X.  N. )
--> Q.
2 addpqf 8852 . . . 4  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
3 fco 5629 . . . 4  |-  ( ( /Q : ( N. 
X.  N. ) --> Q.  /\  +pQ 
: ( ( N. 
X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. ) )  ->  ( /Q  o.  +pQ  ) :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> Q. )
41, 2, 3mp2an 655 . . 3  |-  ( /Q  o.  +pQ  ) :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> Q.
5 elpqn 8833 . . . . 5  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
65ssriv 3338 . . . 4  |-  Q.  C_  ( N.  X.  N. )
7 xpss12 5010 . . . 4  |-  ( ( Q.  C_  ( N.  X.  N. )  /\  Q.  C_  ( N.  X.  N. ) )  ->  ( Q.  X.  Q. )  C_  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) )
86, 6, 7mp2an 655 . . 3  |-  ( Q. 
X.  Q. )  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
9 fssres 5639 . . 3  |-  ( ( ( /Q  o.  +pQ  ) : ( ( N. 
X.  N. )  X.  ( N.  X.  N. ) ) --> Q.  /\  ( Q. 
X.  Q. )  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) ) )  -> 
( ( /Q  o.  +pQ  )  |`  ( Q. 
X.  Q. ) ) : ( Q.  X.  Q. )
--> Q. )
104, 8, 9mp2an 655 . 2  |-  ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) : ( Q.  X.  Q. ) --> Q.
11 df-plq 8822 . . 3  |-  +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q. 
X.  Q. ) )
1211feq1i 5614 . 2  |-  (  +Q  : ( Q.  X.  Q. ) --> Q.  <->  ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) : ( Q.  X.  Q. ) --> Q. )
1310, 12mpbir 202 1  |-  +Q  :
( Q.  X.  Q. )
--> Q.
Colors of variables: wff set class
Syntax hints:    C_ wss 3306    X. cxp 4905    |` cres 4909    o. ccom 4911   -->wf 5479   N.cnpi 8750    +pQ cplpq 8754   Q.cnq 8758   /Qcerq 8760    +Q cplq 8761
This theorem is referenced by:  addcomnq  8859  adderpq  8864  addassnq  8866  distrnq  8869  ltanq  8879  ltexnq  8883  nsmallnq  8885  ltbtwnnq  8886  prlem934  8941  ltaddpr  8942  ltexprlem2  8945  ltexprlem3  8946  ltexprlem4  8947  ltexprlem6  8949  ltexprlem7  8950  prlem936  8955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-omul 6758  df-er 6934  df-ni 8780  df-pli 8781  df-mi 8782  df-lti 8783  df-plpq 8816  df-enq 8819  df-nq 8820  df-erq 8821  df-plq 8822  df-1nq 8824
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