Theorem addlid 8318

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 8171	. . 3 |- 0 e. CC
2		addcom 8316	. . 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 8317	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2266	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202  (class class class)co 6018   CCcc 8030   0cc0 8032   + caddc 8035

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-mulcl 8130  ax-addcom 8132  ax-i2m1 8137  ax-0id 8140

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  readdcan  8319  addlidi  8322  addlidd  8329  cnegexlem1  8354  cnegexlem2  8355  addcan  8359  negneg  8429  fz0to4untppr  10359  fzo0addel  10433  fzoaddel2  10435  divfl0  10556  modqid  10611  swrdspsleq  11248  swrds1  11249  sumrbdclem  11939  summodclem2a  11943  fisum0diag2  12009  eftlub  12252  gcdid  12558  cncrng  14585  ptolemy  15550