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

Theorem mulex 11707
 Description: The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
mulex · ∈ V

Proof of Theorem mulex
StepHypRef Expression
1 ax-mulf 9895 . 2 · :(ℂ × ℂ)⟶ℂ
2 cnex 9896 . . 3 ℂ ∈ V
32, 2xpex 6860 . 2 (ℂ × ℂ) ∈ V
4 fex2 7014 . 2 (( · :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → · ∈ V)
51, 3, 2, 4mp3an 1416 1 · ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Vcvv 3173   × cxp 5036  ⟶wf 5800  ℂcc 9813   · cmul 9820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  cnfldmul  19573  cnlmod4  22747  cnnvg  26917  cnnvs  26919  cncph  27058
 Copyright terms: Public domain W3C validator