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Theorem 2times 9078
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 9009 . . 3 2 = (1 + 1)
21oveq1i 5907 . 2 (2 · 𝐴) = ((1 + 1) · 𝐴)
3 1p1times 8122 . 2 (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴))
42, 3eqtrid 2234 1 (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  (class class class)co 5897  cc 7840  1c1 7843   + caddc 7845   · cmul 7847  2c2 9001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-resscn 7934  ax-1cn 7935  ax-icn 7937  ax-addcl 7938  ax-mulcl 7940  ax-mulcom 7943  ax-mulass 7945  ax-distr 7946  ax-1rid 7949  ax-cnre 7953
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900  df-2 9009
This theorem is referenced by:  times2  9079  2timesi  9080  2halves  9179  halfaddsub  9184  avglt2  9189  2timesd  9192  expubnd  10611  subsq2  10662  sinmul  11787  sin2t  11792  cos2t  11793  pythagtriplem4  12303  pythagtriplem14  12312  pythagtriplem16  12314
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