Theorem addid2i 7251

Description: 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)

Hypothesis

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ

Assertion

Ref	Expression
addid2i	⊢ (0 + 𝐴) = 𝐴

Proof of Theorem addid2i

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		addid2 7247	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 7	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1284   ∈ wcel 1433  (class class class)co 5532  ℂcc 6979  0cc0 6981   + caddc 6984

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-addcom 7076  ax-i2m1 7081  ax-0id 7084

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  ine0  7498  inelr  7684  muleqadd  7758  0p1e1  8153  iap0  8254  num0h  8488  nummul1c  8525  decrmac  8534  decmul1  8540  fz0tp  9135  fzo0to3tp  9228  rei  9786  imi  9787  resqrexlemover  9896  ex-fac  10565