Theorem addlid 8182

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 8035	. . 3 |- 0 e. CC
2		addcom 8180	. . 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 8181	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2231	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   0cc0 7896   + caddc 7899

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-mulcl 7994  ax-addcom 7996  ax-i2m1 8001  ax-0id 8004

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  readdcan  8183  addlidi  8186  addlidd  8193  cnegexlem1  8218  cnegexlem2  8219  addcan  8223  negneg  8293  fz0to4untppr  10216  fzoaddel2  10286  divfl0  10403  modqid  10458  sumrbdclem  11559  summodclem2a  11563  fisum0diag2  11629  eftlub  11872  gcdid  12178  cncrng  14201  ptolemy  15144