Theorem addid2 6949

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 6817	. . 3 ⊢ 0 ∈ ℂ
2		addcom 6947	. . 3 ⊢ ((A ∈ ℂ ∧ 0 ∈ ℂ) → (A + 0) = (0 + A))
3	1, 2	mpan2 401	. 2 ⊢ (A ∈ ℂ → (A + 0) = (0 + A))
4		addid1 6948	. 2 ⊢ (A ∈ ℂ → (A + 0) = A)
5	3, 4	eqtr3d 2071	1 ⊢ (A ∈ ℂ → (0 + A) = A)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  (class class class)co 5455  ℂcc 6709  0cc0 6711   + caddc 6714

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-mulcl 6781  ax-addcom 6783  ax-i2m1 6788  ax-0id 6791

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  readdcan  6950  addid2i  6953  addid2d  6960  cnegexlem1  6983  cnegexlem2  6984  addcan  6988  negneg  7057  fzoaddel2  8819