Theorem 00id 8310

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 8161	. 2 |- 0 e. CC
2		addrid 8307	. 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 6013   CCcc 8020   0cc0 8022   + caddc 8025

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 8115  ax-icn 8117  ax-addcl 8118  ax-mulcl 8120  ax-i2m1 8127  ax-0id 8130

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

This theorem is referenced by:  negdii  8453  addgt0  8618  addgegt0  8619  addgtge0  8620  addge0  8621  add20  8644  recexaplem2  8822  crap0  9128  iap0  9357  decaddm10  9659  10p10e20  9695  ser0  10785  bcpasc  11018  abs00ap  11613  fsumadd  11957  fsumrelem  12022  arisum  12049  bezoutr1  12594  nnnn0modprm0  12818  pcaddlem  12902  4sqlem19  12972  cnfld0  14575  vtxdgfi0e  16101  1kp2ke3k  16256