Theorem addlid 8273

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

Assertion

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

Proof of Theorem addlid

Step	Hyp	Ref	Expression
1		0cn 8126	. . 3 |- 0 e. CC
2		addcom 8271	. . 3 |- ( ( A e. CC /\ 0 e. CC ) -> ( A + 0 ) = ( 0 + A ) )
3	1, 2	mpan2 425	. 2 |- ( A e. CC -> ( A + 0 ) = ( 0 + A ) )
4		addrid 8272	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2264	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 5994   CCcc 7985   0cc0 7987   + caddc 7990

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 8080  ax-icn 8082  ax-addcl 8083  ax-mulcl 8085  ax-addcom 8087  ax-i2m1 8092  ax-0id 8095

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

This theorem is referenced by:  readdcan  8274  addlidi  8277  addlidd  8284  cnegexlem1  8309  cnegexlem2  8310  addcan  8314  negneg  8384  fz0to4untppr  10308  fzo0addel  10381  fzoaddel2  10383  divfl0  10503  modqid  10558  swrdspsleq  11185  swrds1  11186  sumrbdclem  11874  summodclem2a  11878  fisum0diag2  11944  eftlub  12187  gcdid  12493  cncrng  14518  ptolemy  15483