Theorem addlid 8168

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 8021	. . 3 |- 0 e. CC
2		addcom 8166	. . 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 8167	. 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 5923   CCcc 7880   0cc0 7882   + caddc 7885

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 7975  ax-icn 7977  ax-addcl 7978  ax-mulcl 7980  ax-addcom 7982  ax-i2m1 7987  ax-0id 7990

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

This theorem is referenced by:  readdcan  8169  addlidi  8172  addlidd  8179  cnegexlem1  8204  cnegexlem2  8205  addcan  8209  negneg  8279  fz0to4untppr  10202  fzoaddel2  10272  divfl0  10389  modqid  10444  sumrbdclem  11545  summodclem2a  11549  fisum0diag2  11615  eftlub  11858  gcdid  12164  cncrng  14151  ptolemy  15086