Theorem addid2i 8162

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		addlid 8158	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2164  (class class class)co 5918  ℂcc 7870  0cc0 7872   + caddc 7875

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-mulcl 7970  ax-addcom 7972  ax-i2m1 7977  ax-0id 7980

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  ine0  8413  inelr  8603  muleqadd  8687  0p1e1  9096  iap0  9205  num0h  9459  nummul1c  9496  decrmac  9505  decmul1  9511  fz0tp  10188  fz0to4untppr  10190  fzo0to3tp  10286  rei  11043  imi  11044  resqrexlemover  11154  ef01bndlem  11899  efhalfpi  14934  sinq34lt0t  14966  ex-fac  15220