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Theorem mulnqf 8760
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 8741 . . . 4  |-  /Q :
( N.  X.  N. )
--> Q.
2 mulpqf 8757 . . . 4  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
3 fco 5541 . . . 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 8736 . . . . 5  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
65ssriv 3296 . . . 4  |-  Q.  C_  ( N.  X.  N. )
7 xpss12 4922 . . . 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 5551 . . 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-mq 8726 . . 3  |-  .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q. 
X.  Q. ) )
1211feq1i 5526 . 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 3264    X. cxp 4817    |` cres 4821    o. ccom 4823   -->wf 5391   N.cnpi 8653    .pQ cmpq 8658   Q.cnq 8661   /Qcerq 8663    .Q cmq 8665
This theorem is referenced by:  mulcomnq  8764  mulerpq  8768  mulassnq  8770  distrnq  8772  recmulnq  8775  recclnq  8777  dmrecnq  8779  ltmnq  8783  prlem936  8858
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-ni 8683  df-mi 8685  df-lti 8686  df-mpq 8720  df-enq 8722  df-nq 8723  df-erq 8724  df-mq 8726  df-1nq 8727
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