Theorem addlidi 8463

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

Hypothesis

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ

Assertion

Ref	Expression
addlidi	⊢ (0 + 𝐴) = 𝐴

Proof of Theorem addlidi

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		addlid 8459	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1402   ∈ wcel 2209  (class class class)co 6079  ℂcc 8171  0cc0 8173   + caddc 8176

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-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-1cn 8266  ax-icn 8268  ax-addcl 8269  ax-mulcl 8271  ax-addcom 8273  ax-i2m1 8278  ax-0id 8281

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  ine0  8715  inelr  8906  muleqadd  8992  0p1e1  9401  iap0  9511  num0h  9771  nummul1c  9808  decrmac  9817  decmul1  9823  fz0tp  10512  fz0to4untppr  10514  fzo0to3tp  10620  cats1fvn  11519  rei  11648  imi  11649  resqrexlemover  11759  ef01bndlem  12506  5ndvds3  12684  dec5dvds2  13175  2exp11  13198  2exp16  13199  efhalfpi  15883  sinq34lt0t  15915  log2ublem3  16068  log2ublog2  16069  ex-fac  16725