MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addnqf Unicode version

Theorem addnqf 8789
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 8771 . . . 4  |-  /Q :
( N.  X.  N. )
--> Q.
2 addpqf 8785 . . . 4  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
3 fco 5567 . . . 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 654 . . 3  |-  ( /Q  o.  +pQ  ) :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> Q.
5 elpqn 8766 . . . . 5  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
65ssriv 3320 . . . 4  |-  Q.  C_  ( N.  X.  N. )
7 xpss12 4948 . . . 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 654 . . 3  |-  ( Q. 
X.  Q. )  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
9 fssres 5577 . . 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 654 . 2  |-  ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) : ( Q.  X.  Q. ) --> Q.
11 df-plq 8755 . . 3  |-  +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q. 
X.  Q. ) )
1211feq1i 5552 . 2  |-  (  +Q  : ( Q.  X.  Q. ) --> Q.  <->  ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) ) : ( Q.  X.  Q. ) --> Q. )
1310, 12mpbir 201 1  |-  +Q  :
( Q.  X.  Q. )
--> Q.
Colors of variables: wff set class
Syntax hints:    C_ wss 3288    X. cxp 4843    |` cres 4847    o. ccom 4849   -->wf 5417   N.cnpi 8683    +pQ cplpq 8687   Q.cnq 8691   /Qcerq 8693    +Q cplq 8694
This theorem is referenced by:  addcomnq  8792  adderpq  8797  addassnq  8799  distrnq  8802  ltanq  8812  ltexnq  8816  nsmallnq  8818  ltbtwnnq  8819  prlem934  8874  ltaddpr  8875  ltexprlem2  8878  ltexprlem3  8879  ltexprlem4  8880  ltexprlem6  8882  ltexprlem7  8883  prlem936  8888
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6872  df-ni 8713  df-pli 8714  df-mi 8715  df-lti 8716  df-plpq 8749  df-enq 8752  df-nq 8753  df-erq 8754  df-plq 8755  df-1nq 8757
  Copyright terms: Public domain W3C validator