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Theorem mulex 11663
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 9872 . 2 · :(ℂ × ℂ)⟶ℂ
2 cnex 9873 . . 3 ℂ ∈ V
32, 2xpex 6837 . 2 (ℂ × ℂ) ∈ V
4 fex2 6991 . 2 (( · :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → · ∈ V)
51, 3, 2, 4mp3an 1415 1 · ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1976  Vcvv 3172   × cxp 5026  wf 5786  cc 9790   · cmul 9797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-cnv 5036  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by:  cnfldmul  19519  cnlmod4  22676  cnnvg  26713  cnnvs  26716  cncph  26864
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