Theorem addlidi 8297

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

Hypothesis

Ref	Expression
mul.1	⊢ 𝐴 ∈ ℂ

Assertion

Ref	Expression
addlidi	⊢ (0 + 𝐴) = 𝐴

Proof of Theorem addlidi

Step	Hyp	Ref	Expression
1		mul.1	. 2 ⊢ 𝐴 ∈ ℂ
2		addlid 8293	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200  (class class class)co 6007  ℂcc 8005  0cc0 8007   + caddc 8010

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-1cn 8100  ax-icn 8102  ax-addcl 8103  ax-mulcl 8105  ax-addcom 8107  ax-i2m1 8112  ax-0id 8115

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  ine0  8548  inelr  8739  muleqadd  8823  0p1e1  9232  iap0  9342  num0h  9597  nummul1c  9634  decrmac  9643  decmul1  9649  fz0tp  10326  fz0to4untppr  10328  fzo0to3tp  10433  cats1fvn  11304  rei  11418  imi  11419  resqrexlemover  11529  ef01bndlem  12275  5ndvds3  12453  dec5dvds2  12944  2exp11  12967  2exp16  12968  efhalfpi  15481  sinq34lt0t  15513  ex-fac  16116