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Theorem times2d 11118
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 10988 . 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 1474  wcel 1975  (class class class)co 6522  cc 9785   + caddc 9790   · cmul 9792  2c2 10912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rrecex 9859  ax-cnre 9860
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 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-iota 5749  df-fv 5793  df-ov 6525  df-2 10921
This theorem is referenced by:  div4p1lem1div2  11129  climcndslem1  14361  climcndslem2  14362  sadcaddlem  14958  dvexp3  23457  chordthmlem  24271  chordthmlem2  24272  chordthmlem4  24274  logfaclbnd  24659  rplogsumlem1  24885  nexple  29200
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