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

Theorem mulex 10611
Description: The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
mulex  |-  x.  e.  _V

Proof of Theorem mulex
StepHypRef Expression
1 ax-mulf 9070 . 2  |-  x.  :
( CC  X.  CC )
--> CC
2 cnex 9071 . . 3  |-  CC  e.  _V
32, 2xpex 4990 . 2  |-  ( CC 
X.  CC )  e. 
_V
4 fex2 5603 . 2  |-  ( (  x.  : ( CC 
X.  CC ) --> CC 
/\  ( CC  X.  CC )  e.  _V  /\  CC  e.  _V )  ->  x.  e.  _V )
51, 3, 2, 4mp3an 1279 1  |-  x.  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956    X. cxp 4876   -->wf 5450   CCcc 8988    x. cmul 8995
This theorem is referenced by:  cnfldmul  16709  cnrngo  21991  cnnvg  22169  cnnvs  22172  cncph  22320
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458
  Copyright terms: Public domain W3C validator