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Theorem times2d 12494
Description: A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
Hypothesis
Ref Expression
2timesd.1 (𝜑𝐴 ∈ ℂ)
Assertion
Ref Expression
times2d (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))

Proof of Theorem times2d
StepHypRef Expression
1 2timesd.1 . 2 (𝜑𝐴 ∈ ℂ)
2 times2 12387 . 2 (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
31, 2syl 17 1 (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  (class class class)co 7419  cc 11143   + caddc 11148   · cmul 11150  2c2 12305
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-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-mulcl 11207  ax-mulcom 11209  ax-mulass 11211  ax-distr 11212  ax-1rid 11215  ax-cnre 11218
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-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-2 12313
This theorem is referenced by:  div4p1lem1div2  12505  climcndslem1  15836  climcndslem2  15837  sadcaddlem  16440  dvexp3  25959  chordthmlem  26814  chordthmlem2  26815  chordthmlem4  26817  logfaclbnd  27205  rplogsumlem1  27467  nexple  33761  aks4d1p1p5  41680  fltne  42205
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