ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2times GIF version

Theorem 2times 9006
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 8937 . . 3 2 = (1 + 1)
21oveq1i 5863 . 2 (2 · 𝐴) = ((1 + 1) · 𝐴)
3 1p1times 8053 . 2 (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴))
42, 3eqtrid 2215 1 (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  (class class class)co 5853  cc 7772  1c1 7775   + caddc 7777   · cmul 7779  2c2 8929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-mulcl 7872  ax-mulcom 7875  ax-mulass 7877  ax-distr 7878  ax-1rid 7881  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856  df-2 8937
This theorem is referenced by:  times2  9007  2timesi  9008  2halves  9107  halfaddsub  9112  avglt2  9117  2timesd  9120  expubnd  10533  subsq2  10583  sinmul  11707  sin2t  11712  cos2t  11713  pythagtriplem4  12222  pythagtriplem14  12231  pythagtriplem16  12233
  Copyright terms: Public domain W3C validator