Theorem addid2 7152

Description: 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
addid2	|- ( A e. CC -> ( 0 + A ) = A )

Proof of Theorem addid2

Step	Hyp	Ref	Expression
1		0cn 7019	. . 3 |- 0 e. CC
2		addcom 7150	. . 3 |- ( ( A e. CC /\ 0 e. CC ) -> ( A + 0 ) = ( 0 + A ) )
3	1, 2	mpan2 401	. 2 |- ( A e. CC -> ( A + 0 ) = ( 0 + A ) )
4		addid1 7151	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2074	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 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-addcom 6984  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:  readdcan  7153  addid2i  7156  addid2d  7163  cnegexlem1  7186  cnegexlem2  7187  addcan  7191  negneg  7261  fzoaddel2  9049  divfl0  9138