Theorem addid2 7247

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 7111	. . 3 |- 0 e. CC
2		addcom 7245	. . 3 |- ( ( A e. CC /\ 0 e. CC ) -> ( A + 0 ) = ( 0 + A ) )
3	1, 2	mpan2 415	. 2 |- ( A e. CC -> ( A + 0 ) = ( 0 + A ) )
4		addid1 7246	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2115	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   0cc0 6981   + caddc 6984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-addcom 7076  ax-i2m1 7081  ax-0id 7084

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  readdcan  7248  addid2i  7251  addid2d  7258  cnegexlem1  7283  cnegexlem2  7284  addcan  7288  negneg  7358  fzoaddel2  9202  divfl0  9298  modqid  9351  gcdid  10377