Theorem 00id 8313

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 8164	. 2 ⊢ 0 ∈ ℂ
2		addrid 8310	. 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 6013  ℂcc 8023  0cc0 8025   + caddc 8028

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 8118  ax-icn 8120  ax-addcl 8121  ax-mulcl 8123  ax-i2m1 8130  ax-0id 8133

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

This theorem is referenced by:  negdii  8456  addgt0  8621  addgegt0  8622  addgtge0  8623  addge0  8624  add20  8647  recexaplem2  8825  crap0  9131  iap0  9360  decaddm10  9662  10p10e20  9698  ser0  10788  bcpasc  11021  abs00ap  11616  fsumadd  11960  fsumrelem  12025  arisum  12052  bezoutr1  12597  nnnn0modprm0  12821  pcaddlem  12905  4sqlem19  12975  cnfld0  14578  vtxdgfi0e  16106  1kp2ke3k  16270