Theorem addlidi 8315

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 8311	. 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 6013  ℂcc 8023  0cc0 8025   + caddc 8028

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 8118  ax-icn 8120  ax-addcl 8121  ax-mulcl 8123  ax-addcom 8125  ax-i2m1 8130  ax-0id 8133

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

This theorem is referenced by:  ine0  8566  inelr  8757  muleqadd  8841  0p1e1  9250  iap0  9360  num0h  9615  nummul1c  9652  decrmac  9661  decmul1  9667  fz0tp  10350  fz0to4untppr  10352  fzo0to3tp  10457  cats1fvn  11338  rei  11453  imi  11454  resqrexlemover  11564  ef01bndlem  12310  5ndvds3  12488  dec5dvds2  12979  2exp11  13002  2exp16  13003  efhalfpi  15516  sinq34lt0t  15548  ex-fac  16274