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Theorem mulex 12367
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 10595 . 2 · :(ℂ × ℂ)⟶ℂ
2 cnex 10596 . . 3 ℂ ∈ V
32, 2xpex 7454 . 2 (ℂ × ℂ) ∈ V
4 fex2 7616 . 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 2114  Vcvv 3473   × cxp 5529  wf 6327  cc 10513   · cmul 10520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-mulf 10595
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-dm 5541  df-rn 5542  df-fun 6333  df-fn 6334  df-f 6335
This theorem is referenced by:  cnfldmul  20527  cnfldfun  20533  cnfldfunALT  20534  cnlmod4  23723  cnnvg  28440  cnnvs  28442  cncph  28581
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