Theorem addlid 8361

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 8214	. . 3 |- 0 e. CC
2		addcom 8359	. . 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 8360	. 2 |- ( A e. CC -> ( A + 0 ) = A )
5	3, 4	eqtr3d 2266	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8073   0cc0 8075   + caddc 8078

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 2213  ax-1cn 8168  ax-icn 8170  ax-addcl 8171  ax-mulcl 8173  ax-addcom 8175  ax-i2m1 8180  ax-0id 8183

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

This theorem is referenced by:  readdcan  8362  addlidi  8365  addlidd  8372  cnegexlem1  8397  cnegexlem2  8398  addcan  8402  negneg  8472  fz0to4untppr  10402  fzo0addel  10477  fzoaddel2  10479  divfl0  10600  modqid  10655  swrdspsleq  11295  swrds1  11296  sumrbdclem  11999  summodclem2a  12003  fisum0diag2  12069  eftlub  12312  gcdid  12618  cncrng  14645  ptolemy  15615