Theorem addlid 8411

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 8265	. . 3 |- 0 e. CC
2		addcom 8409	. . 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 8410	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2267	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203  (class class class)co 6049   CCcc 8124   0cc0 8126   + caddc 8129

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-1cn 8219  ax-icn 8221  ax-addcl 8222  ax-mulcl 8224  ax-addcom 8226  ax-i2m1 8231  ax-0id 8234

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  readdcan  8412  addlidi  8415  addlidd  8422  cnegexlem1  8447  cnegexlem2  8448  addcan  8452  negneg  8522  fz0to4untppr  10457  fzo0addel  10532  fzoaddel2  10534  divfl0  10655  modqid  10710  swrdspsleq  11355  swrds1  11356  sumrbdclem  12059  summodclem2a  12063  fisum0diag2  12129  eftlub  12372  gcdid  12678  cncrng  14709  ptolemy  15681