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Theorem mulnqf 8589
Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
mulnqf  |-  .Q  :
( Q.  X.  Q. )
--> Q.

Proof of Theorem mulnqf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nqerf 8570 . . . 4  |-  /Q :
( N.  X.  N. )
--> Q.
2 mulpqf 8586 . . . 4  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
3 fco 5414 . . . 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 653 . . 3  |-  ( /Q  o.  .pQ  ) :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> Q.
5 elpqn 8565 . . . . 5  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
65ssriv 3197 . . . 4  |-  Q.  C_  ( N.  X.  N. )
7 xpss12 4808 . . . 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 653 . . 3  |-  ( Q. 
X.  Q. )  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
9 fssres 5424 . . 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 653 . 2  |-  ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) : ( Q.  X.  Q. ) --> Q.
11 df-mq 8555 . . 3  |-  .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q. 
X.  Q. ) )
1211feq1i 5399 . 2  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  <->  ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) ) : ( Q.  X.  Q. ) --> Q. )
1310, 12mpbir 200 1  |-  .Q  :
( Q.  X.  Q. )
--> Q.
Colors of variables: wff set class
Syntax hints:    C_ wss 3165    X. cxp 4703    |` cres 4707    o. ccom 4709   -->wf 5267   N.cnpi 8482    .pQ cmpq 8487   Q.cnq 8490   /Qcerq 8492    .Q cmq 8494
This theorem is referenced by:  mulcomnq  8593  mulerpq  8597  mulassnq  8599  distrnq  8601  recmulnq  8604  recclnq  8606  dmrecnq  8608  ltmnq  8612  prlem936  8687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-mi 8514  df-lti 8515  df-mpq 8549  df-enq 8551  df-nq 8552  df-erq 8553  df-mq 8555  df-1nq 8556
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