ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addid2d GIF version

Theorem addid2d 7224
Description: 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
Hypothesis
Ref Expression
muld.1 (𝜑𝐴 ∈ ℂ)
Assertion
Ref Expression
addid2d (𝜑 → (0 + 𝐴) = 𝐴)

Proof of Theorem addid2d
StepHypRef Expression
1 muld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addid2 7213 . 2 (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
31, 2syl 14 1 (𝜑 → (0 + 𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  (class class class)co 5540  cc 6945  0cc0 6947   + caddc 6950
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-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-mulcl 7040  ax-addcom 7042  ax-i2m1 7047  ax-0id 7050
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  negeu  7265  ltadd2  7488  subge0  7544  sublt0d  7635  un0addcl  8272  lincmb01cmp  8972  modsumfzodifsn  9346  rennim  9829  moddvds  10117
  Copyright terms: Public domain W3C validator