Theorem addid2 7925

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 7782	. . 3 ⊢ 0 ∈ ℂ
2		addcom 7923	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 + 0) = (0 + 𝐴))
3	1, 2	mpan2 422	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = (0 + 𝐴))
4		addid1 7924	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
5	3, 4	eqtr3d 2175	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  (class class class)co 5782  ℂcc 7642  0cc0 7644   + caddc 7647

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-mulcl 7742  ax-addcom 7744  ax-i2m1 7749  ax-0id 7752

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136

This theorem is referenced by:  readdcan  7926  addid2i  7929  addid2d  7936  cnegexlem1  7961  cnegexlem2  7962  addcan  7966  negneg  8036  fzoaddel2  10001  divfl0  10100  modqid  10153  sumrbdclem  11178  summodclem2a  11182  fisum0diag2  11248  eftlub  11433  gcdid  11710  ptolemy  12953