Theorem 00id 8283

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 8134	. 2 |- 0 e. CC
2		addrid 8280	. 2 |- ( 0 e. CC -> ( 0 + 0 ) = 0 )
3	1, 2	ax-mp 5	1 |- ( 0 + 0 ) = 0

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6000   CCcc 7993   0cc0 7995   + caddc 7998

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 8088  ax-icn 8090  ax-addcl 8091  ax-mulcl 8093  ax-i2m1 8100  ax-0id 8103

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

This theorem is referenced by:  negdii  8426  addgt0  8591  addgegt0  8592  addgtge0  8593  addge0  8594  add20  8617  recexaplem2  8795  crap0  9101  iap0  9330  decaddm10  9632  10p10e20  9668  ser0  10750  bcpasc  10983  abs00ap  11568  fsumadd  11912  fsumrelem  11977  arisum  12004  bezoutr1  12549  nnnn0modprm0  12773  pcaddlem  12857  4sqlem19  12927  cnfld0  14529  1kp2ke3k  16046