Theorem addlidi 8322

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 8318	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

Syntax hints:   = wceq 1397   ∈ wcel 2202  (class class class)co 6018  ℂcc 8030  0cc0 8032   + caddc 8035

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-mulcl 8130  ax-addcom 8132  ax-i2m1 8137  ax-0id 8140

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  ine0  8573  inelr  8764  muleqadd  8848  0p1e1  9257  iap0  9367  num0h  9622  nummul1c  9659  decrmac  9668  decmul1  9674  fz0tp  10357  fz0to4untppr  10359  fzo0to3tp  10465  cats1fvn  11346  rei  11461  imi  11462  resqrexlemover  11572  ef01bndlem  12319  5ndvds3  12497  dec5dvds2  12988  2exp11  13011  2exp16  13012  efhalfpi  15526  sinq34lt0t  15558  ex-fac  16341