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

Theorem mulpqnq 8810
 Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
Assertion
Ref Expression
mulpqnq

Proof of Theorem mulpqnq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mq 8784 . . . . 5
21fveq1i 5721 . . . 4
32a1i 11 . . 3
4 opelxpi 4902 . . . 4
5 fvres 5737 . . . 4
64, 5syl 16 . . 3
7 df-mpq 8778 . . . . 5
8 opex 4419 . . . . 5
97, 8fnmpt2i 6412 . . . 4
10 elpqn 8794 . . . . 5
11 elpqn 8794 . . . . 5
12 opelxpi 4902 . . . . 5
1310, 11, 12syl2an 464 . . . 4
14 fvco2 5790 . . . 4
159, 13, 14sylancr 645 . . 3
163, 6, 153eqtrd 2471 . 2
17 df-ov 6076 . 2
18 df-ov 6076 . . 3
1918fveq2i 5723 . 2
2016, 17, 193eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cop 3809   cxp 4868   cres 4872   ccom 4874   wfn 5441  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340  cnpi 8711   cmi 8713   cmpq 8716  cnq 8719  cerq 8721   cmq 8723 This theorem is referenced by:  mulclnq  8816  mulcomnq  8822  mulerpq  8826  mulassnq  8828  distrnq  8830  mulidnq  8832  ltmnq  8841 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-mpq 8778  df-nq 8781  df-mq 8784
 Copyright terms: Public domain W3C validator