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Theorem 2times 8111
Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
Assertion
Ref Expression
2times (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))

Proof of Theorem 2times
StepHypRef Expression
1 df-2 8049 . . 3 2 = (1 + 1)
21oveq1i 5550 . 2 (2 · 𝐴) = ((1 + 1) · 𝐴)
3 1p1times 7208 . 2 (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴))
42, 3syl5eq 2100 1 (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  (class class class)co 5540  cc 6945  1c1 6948   + caddc 6950   · cmul 6952  2c2 8040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-mulcl 7040  ax-mulcom 7043  ax-mulass 7045  ax-distr 7046  ax-1rid 7049  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543  df-2 8049
This theorem is referenced by:  times2  8112  2timesi  8113  2halves  8211  halfaddsub  8216  avglt2  8221  2timesd  8224  expubnd  9477  subsq2  9526
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