Theorem 00id 8298

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 8149	. 2 |- 0 e. CC
2		addrid 8295	. 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 6007   CCcc 8008   0cc0 8010   + caddc 8013

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 8103  ax-icn 8105  ax-addcl 8106  ax-mulcl 8108  ax-i2m1 8115  ax-0id 8118

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

This theorem is referenced by:  negdii  8441  addgt0  8606  addgegt0  8607  addgtge0  8608  addge0  8609  add20  8632  recexaplem2  8810  crap0  9116  iap0  9345  decaddm10  9647  10p10e20  9683  ser0  10767  bcpasc  11000  abs00ap  11588  fsumadd  11932  fsumrelem  11997  arisum  12024  bezoutr1  12569  nnnn0modprm0  12793  pcaddlem  12877  4sqlem19  12947  cnfld0  14550  vtxdgfi0e  16054  1kp2ke3k  16143