Theorem addid2i 8148

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 8144	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2160  (class class class)co 5906  ℂcc 7856  0cc0 7858   + caddc 7861

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 2171  ax-1cn 7951  ax-icn 7953  ax-addcl 7954  ax-mulcl 7956  ax-addcom 7958  ax-i2m1 7963  ax-0id 7966

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  ine0  8399  inelr  8589  muleqadd  8673  0p1e1  9082  iap0  9191  num0h  9445  nummul1c  9482  decrmac  9491  decmul1  9497  fz0tp  10174  fz0to4untppr  10176  fzo0to3tp  10272  rei  11017  imi  11018  resqrexlemover  11128  ef01bndlem  11873  efhalfpi  14862  sinq34lt0t  14894  ex-fac  15144