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Theorem addid2 7303
Description: 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
addid2 (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Proof of Theorem addid2
StepHypRef Expression
1 0cn 7162 . . 3 0 ∈ ℂ
2 addcom 7301 . . 3 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
31, 2mpan2 416 . 2 (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4 addid1 7302 . 2 (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
53, 4eqtr3d 2116 1 (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  (class class class)co 5537  cc 7030  0cc0 7032   + caddc 7035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064  ax-1cn 7120  ax-icn 7122  ax-addcl 7123  ax-mulcl 7125  ax-addcom 7127  ax-i2m1 7132  ax-0id 7135
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078
This theorem is referenced by:  readdcan  7304  addid2i  7307  addid2d  7314  cnegexlem1  7339  cnegexlem2  7340  addcan  7344  negneg  7414  fzoaddel2  9268  divfl0  9367  modqid  9420  isumrblem  10326  gcdid  10510
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