Theorem addassi 8142

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 8117	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	mp3an 1371	1 |- ( ( A + B ) + C ) = ( A + ( B + C ) )

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 5994   CCcc 7985   + caddc 7990

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 8089

This theorem depends on definitions:  df-bi 117  df-3an 1004

This theorem is referenced by:  2p2e4  9225  3p2e5  9240  3p3e6  9241  4p2e6  9242  4p3e7  9243  4p4e8  9244  5p2e7  9245  5p3e8  9246  5p4e9  9247  6p2e8  9248  6p3e9  9249  7p2e9  9250  numsuc  9579  nummac  9610  numaddc  9613  6p5lem  9635  5p5e10  9636  6p4e10  9637  7p3e10  9640  8p2e10  9645  binom2i  10857  resqrexlemover  11507  3dvdsdec  12362  3dvds2dec  12363  decsplit  12938  lgsdir2lem2  15693  2lgsoddprmlem3d  15774