Theorem addid2i 8041

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

Hypothesis

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ

Assertion

Ref	Expression
addid2i	⊢ (0 + 𝐴) = 𝐴

Proof of Theorem addid2i

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		addid2 8037	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1343   ∈ wcel 2136  (class class class)co 5842  ℂcc 7751  0cc0 7753   + caddc 7756

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 7846  ax-icn 7848  ax-addcl 7849  ax-mulcl 7851  ax-addcom 7853  ax-i2m1 7858  ax-0id 7861

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

This theorem is referenced by:  ine0  8292  inelr  8482  muleqadd  8565  0p1e1  8971  iap0  9080  num0h  9333  nummul1c  9370  decrmac  9379  decmul1  9385  fz0tp  10057  fz0to4untppr  10059  fzo0to3tp  10154  rei  10841  imi  10842  resqrexlemover  10952  ef01bndlem  11697  efhalfpi  13360  sinq34lt0t  13392  ex-fac  13609