Theorem addassi 8230

Description: Associative law for addition. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	|- A e. CC
axi.2	|- B e. CC
axi.3	|- C e. CC

Assertion

Ref	Expression
addassi	|- ( ( A + B ) + C ) = ( A + ( B + C ) )

Proof of Theorem addassi

Step	Hyp	Ref	Expression
1		axi.1	. 2 |- A e. CC
2		axi.2	. 2 |- B e. CC
3		axi.3	. 2 |- C e. CC
4		addass 8205	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	mp3an 1374	1 |- ( ( A + B ) + C ) = ( A + ( B + C ) )

Syntax hints:   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8073   + 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-addass 8177

This theorem depends on definitions:  df-bi 117  df-3an 1007

This theorem is referenced by:  2p2e4  9313  3p2e5  9328  3p3e6  9329  4p2e6  9330  4p3e7  9331  4p4e8  9332  5p2e7  9333  5p3e8  9334  5p4e9  9335  6p2e8  9336  6p3e9  9337  7p2e9  9338  numsuc  9667  nummac  9698  numaddc  9701  6p5lem  9723  5p5e10  9724  6p4e10  9725  7p3e10  9728  8p2e10  9733  binom2i  10954  resqrexlemover  11631  3dvdsdec  12487  3dvds2dec  12488  decsplit  13063  lgsdir2lem2  15828  2lgsoddprmlem3d  15909