Theorem 00id 8295

Description: 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
00id	⊢ (0 + 0) = 0

Proof of Theorem 00id

Step	Hyp	Ref	Expression
1		0cn 8146	. 2 ⊢ 0 ∈ ℂ
2		addrid 8292	. 2 ⊢ (0 ∈ ℂ → (0 + 0) = 0)
3	1, 2	ax-mp 5	1 ⊢ (0 + 0) = 0

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-i2m1 8112  ax-0id 8115

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

This theorem is referenced by:  negdii  8438  addgt0  8603  addgegt0  8604  addgtge0  8605  addge0  8606  add20  8629  recexaplem2  8807  crap0  9113  iap0  9342  decaddm10  9644  10p10e20  9680  ser0  10763  bcpasc  10996  abs00ap  11581  fsumadd  11925  fsumrelem  11990  arisum  12017  bezoutr1  12562  nnnn0modprm0  12786  pcaddlem  12870  4sqlem19  12940  cnfld0  14543  1kp2ke3k  16112