Theorem addassi 8334

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 8309	. 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 6085   CCcc 8177   + caddc 8182

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 8281

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

This theorem is used by:  2p2e4  9432  3p2e5  9447  3p3e6  9448  4p2e6  9449  4p3e7  9450  4p4e8  9451  5p2e7  9452  5p3e8  9453  5p4e9  9454  6p2e8  9455  6p3e9  9456  7p2e9  9457  numsuc  9792  nummac  9823  numaddc  9826  6p5lem  9848  5p5e10  9849  6p4e10  9850  7p3e10  9853  8p2e10  9858  binom2i  11087  resqrexlemover  11778  3dvdsdec  12634  3dvds2dec  12635  decsplit  13210  lgsdir2lem2  16160  2lgsoddprmlem3d  16241