Theorem 00id 8303

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 8154	. 2 ⊢ 0 ∈ ℂ
2		addrid 8300	. 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 6010  ℂcc 8013  0cc0 8015   + caddc 8018

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 8108  ax-icn 8110  ax-addcl 8111  ax-mulcl 8113  ax-i2m1 8120  ax-0id 8123

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

This theorem is referenced by:  negdii  8446  addgt0  8611  addgegt0  8612  addgtge0  8613  addge0  8614  add20  8637  recexaplem2  8815  crap0  9121  iap0  9350  decaddm10  9652  10p10e20  9688  ser0  10772  bcpasc  11005  abs00ap  11594  fsumadd  11938  fsumrelem  12003  arisum  12030  bezoutr1  12575  nnnn0modprm0  12799  pcaddlem  12883  4sqlem19  12953  cnfld0  14556  vtxdgfi0e  16081  1kp2ke3k  16197