Theorem addassd 8180

Description: Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
addcld.1	|- ( ph -> A e. CC )
addcld.2	|- ( ph -> B e. CC )
addassd.3	|- ( ph -> C e. CC )

Assertion

Ref	Expression
addassd	|- ( ph -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )

Proof of Theorem addassd

Step	Hyp	Ref	Expression
1		addcld.1	. 2 |- ( ph -> A e. CC )
2		addcld.2	. 2 |- ( ph -> B e. CC )
3		addassd.3	. 2 |- ( ph -> C e. CC )
4		addass 8140	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	syl3anc 1271	1 |- ( ph -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 6007   CCcc 8008   + caddc 8013

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 8112

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

This theorem is referenced by:  readdcan  8297  muladd11r  8313  cnegexlem1  8332  cnegex  8335  addcan  8337  addcan2  8338  negeu  8348  addsubass  8367  nppcan3  8381  muladd  8541  ltadd2  8577  add1p1  9372  div4p1lem1div2  9376  peano2z  9493  zaddcllempos  9494  zpnn0elfzo1  10426  exbtwnzlemstep  10479  rebtwn2zlemstep  10484  flhalf  10534  flqdiv  10555  binom2  10885  binom3  10891  bernneq  10894  omgadd  11036  ccatass  11156  cvg1nlemres  11511  recvguniqlem  11520  resqrexlemover  11536  bdtrilem  11765  bdtri  11766  bcxmas  12015  efsep  12217  efi4p  12243  efival  12258  divalglemnqt  12446  flodddiv4  12462  gcdaddm  12520  pcadd2  12879  4sqlem11  12939  limcimolemlt  15353  tangtx  15527  binom4  15668  2lgslem3c  15789  2lgslem3d  15790