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Theorem addasssr 8898
Description: Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
addasssr  |-  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) )

Proof of Theorem addasssr
Dummy variables  u  v  w  x  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 8870 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 8885 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
x  +P.  z ) ,  ( y  +P.  w ) >. ]  ~R  )
3 addsrpr 8885 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  +R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
z  +P.  v ) ,  ( w  +P.  u ) >. ]  ~R  )
4 addsrpr 8885 . . 3  |-  ( ( ( ( x  +P.  z )  e.  P.  /\  ( y  +P.  w
)  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. ( x  +P.  z
) ,  ( y  +P.  w ) >. ]  ~R  +R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( x  +P.  z
)  +P.  v ) ,  ( ( y  +P.  w )  +P.  u ) >. ]  ~R  )
5 addsrpr 8885 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( z  +P.  v )  e.  P.  /\  ( w  +P.  u
)  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  +R  [ <. ( z  +P.  v ) ,  ( w  +P.  u ) >. ]  ~R  )  =  [ <. (
x  +P.  ( z  +P.  v ) ) ,  ( y  +P.  (
w  +P.  u )
) >. ]  ~R  )
6 addclpr 8830 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  +P.  z
)  e.  P. )
7 addclpr 8830 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  +P.  w
)  e.  P. )
86, 7anim12i 550 . . . 4  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  z )  e.  P.  /\  ( y  +P.  w )  e. 
P. ) )
98an4s 800 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  z )  e.  P.  /\  ( y  +P.  w )  e. 
P. ) )
10 addclpr 8830 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
11 addclpr 8830 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
1210, 11anim12i 550 . . . 4  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
1312an4s 800 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
14 addasspr 8834 . . 3  |-  ( ( x  +P.  z )  +P.  v )  =  ( x  +P.  (
z  +P.  v )
)
15 addasspr 8834 . . 3  |-  ( ( y  +P.  w )  +P.  u )  =  ( y  +P.  (
w  +P.  u )
)
161, 2, 3, 4, 5, 9, 13, 14, 15ecovass 6954 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  +R  B
)  +R  C )  =  ( A  +R  ( B  +R  C
) ) )
17 dmaddsr 8895 . . 3  |-  dom  +R  =  ( R.  X.  R. )
18 0nsr 8889 . . 3  |-  -.  (/)  e.  R.
1917, 18ndmovass 6176 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  +R  B )  +R  C
)  =  ( A  +R  ( B  +R  C ) ) )
2016, 19pm2.61i 158 1  |-  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6022   P.cnp 8669    +P. cpp 8671    ~R cer 8676   R.cnr 8677    +R cplr 8681
This theorem is referenced by:  map2psrpr  8920  axaddass  8966  axmulass  8967  axdistr  8968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-omul 6667  df-er 6843  df-ec 6845  df-qs 6849  df-ni 8684  df-pli 8685  df-mi 8686  df-lti 8687  df-plpq 8720  df-mpq 8721  df-ltpq 8722  df-enq 8723  df-nq 8724  df-erq 8725  df-plq 8726  df-mq 8727  df-1nq 8728  df-rq 8729  df-ltnq 8730  df-np 8793  df-plp 8795  df-ltp 8797  df-plpr 8867  df-enr 8869  df-nr 8870  df-plr 8871
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