Theorem addid2 8033

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

Step	Hyp	Ref	Expression
1		0cn 7887	. . 3 ⊢ 0 ∈ ℂ
2		addcom 8031	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 422	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addid1 8032	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2200	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  (class class class)co 5841  ℂcc 7747  0cc0 7749   + caddc 7752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-1cn 7842  ax-icn 7844  ax-addcl 7845  ax-mulcl 7847  ax-addcom 7849  ax-i2m1 7854  ax-0id 7857

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161

This theorem is referenced by:  readdcan  8034  addid2i  8037  addid2d  8044  cnegexlem1  8069  cnegexlem2  8070  addcan  8074  negneg  8144  fz0to4untppr  10055  fzoaddel2  10124  divfl0  10227  modqid  10280  sumrbdclem  11314  summodclem2a  11318  fisum0diag2  11384  eftlub  11627  gcdid  11915  ptolemy  13345