Theorem 00id 7154

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 7019	. 2 ⊢ 0 ∈ ℂ
2		addid1 7151	. 2 ⊢ (0 ∈ ℂ → (0 + 0) = 0)
3	1, 2	ax-mp 7	1 ⊢ (0 + 0) = 0

Syntax hints:   = wceq 1243   ∈ wcel 1393  (class class class)co 5512  ℂcc 6887  0cc0 6889   + caddc 6892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-mulcl 6982  ax-i2m1 6989  ax-0id 6992

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  negdii  7295  addgt0  7443  addgegt0  7444  addgtge0  7445  addge0  7446  add20  7469  recexaplem2  7633  crap0  7910  iap0  8148  10p10e20  8437  iser0  9250  abs00ap  9660