Theorem addlid 8459

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

Assertion

Ref	Expression
addlid	⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Proof of Theorem addlid

Step	Hyp	Ref	Expression
1		0cn 8312	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8457	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 429	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addrid 8458	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2273	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = 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:  readdcan  8460  addlidi  8463  addlidd  8470  cnegexlem1  8495  cnegexlem2  8496  addcan  8500  negneg  8570  fz0to4untppr  10514  fzo0addel  10589  fzoaddel2  10591  divfl0  10714  modqid  10769  swrdspsleq  11422  swrds1  11423  sumrbdclem  12127  summodclem2a  12131  fisum0diag2  12197  eftlub  12440  gcdid  12746  cncrng  14889  ptolemy  15908