Theorem addlidi 8469

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 8465	. 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (0 + 𝐴) = 𝐴

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  (class class class)co 6085  ℂcc 8177  0cc0 8179   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-mulcl 8277  ax-addcom 8279  ax-i2m1 8284  ax-0id 8287

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  ine0  8721  inelr  8913  muleqadd  8999  0p1e1  9419  iap0  9530  num0h  9790  nummul1c  9827  decrmac  9836  decmul1  9842  fz0tp  10531  fz0to4untppr  10533  fzo0to3tp  10639  cats1fvn  11538  rei  11667  imi  11668  resqrexlemover  11778  ef01bndlem  12525  5ndvds3  12703  dec5dvds2  13194  2exp11  13217  2exp16  13218  efhalfpi  15903  sinq34lt0t  15935  log2ublem3  16091  log2ublog2  16092  ex-fac  16754