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Theorem mulex 13022
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 11234 . 2 · :(ℂ × ℂ)⟶ℂ
2 cnex 11235 . . 3 ℂ ∈ V
32, 2xpex 7760 . 2 (ℂ × ℂ) ∈ V
4 fex2 7946 . 2 (( · :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → · ∈ V)
51, 3, 2, 4mp3an 1457 1 · ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3461   × cxp 5679  wf 6549  cc 11152   · cmul 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-mulf 11234
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5687  df-rel 5688  df-cnv 5689  df-dm 5691  df-rn 5692  df-fun 6555  df-fn 6556  df-f 6557
This theorem is referenced by:  cnfldmulOLD  21356  cnfldfunOLD  21362  cnfldfunALTOLD  21363  cnfldfunALTOLDOLD  21364  cnlmod4  25149  cnnvg  30603  cnnvs  30605  cncph  30744
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