Theorem addassi 8341

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 8316	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	mp3an 1378	1 |- ( ( A + B ) + C ) = ( A + ( B + C ) )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8184   + caddc 8189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-addass 8288

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  2p2e4  9441  3p2e5  9456  3p3e6  9457  4p2e6  9458  4p3e7  9459  4p4e8  9460  5p2e7  9461  5p3e8  9462  5p4e9  9463  6p2e8  9464  6p3e9  9465  7p2e9  9466  numsuc  9802  nummac  9838  numaddc  9841  6p5lem  9863  5p5e10  9864  6p4e10  9865  7p3e10  9868  8p2e10  9873  binom2i  11107  resqrexlemover  11799  3dvdsdec  12658  3dvds2dec  12659  mod2xnegi  13228  decsplit  13239  lgsdir2lem2  16365  2lgsoddprmlem3d  16446