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Theorem mulex 12973
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 11190 . 2 · :(ℂ × ℂ)⟶ℂ
2 cnex 11191 . . 3 ℂ ∈ V
32, 2xpex 7740 . 2 (ℂ × ℂ) ∈ V
4 fex2 7924 . 2 (( · :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → · ∈ V)
51, 3, 2, 4mp3an 1462 1 · ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3475   × cxp 5675  wf 6540  cc 11108   · cmul 11115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  cnfldmul  20950  cnfldfun  20956  cnfldfunALT  20957  cnfldfunALTOLD  20958  cnlmod4  24655  cnnvg  29931  cnnvs  29933  cncph  30072
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