Theorem addid2i 8098

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 8094	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1353   ∈ wcel 2148  (class class class)co 5874  ℂcc 7808  0cc0 7810   + caddc 7813

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-mulcl 7908  ax-addcom 7910  ax-i2m1 7915  ax-0id 7918

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  ine0  8349  inelr  8539  muleqadd  8623  0p1e1  9031  iap0  9140  num0h  9393  nummul1c  9430  decrmac  9439  decmul1  9445  fz0tp  10119  fz0to4untppr  10121  fzo0to3tp  10216  rei  10903  imi  10904  resqrexlemover  11014  ef01bndlem  11759  efhalfpi  14113  sinq34lt0t  14145  ex-fac  14362