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Theorem addasssr 8919
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 8891 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 8906 . . 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 8906 . . 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 8906 . . 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 8906 . . 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 8851 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  +P.  z
)  e.  P. )
7 addclpr 8851 . . . . 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 8851 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
11 addclpr 8851 . . . . 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 8855 . . 3  |-  ( ( x  +P.  z )  +P.  v )  =  ( x  +P.  (
z  +P.  v )
)
15 addasspr 8855 . . 3  |-  ( ( y  +P.  w )  +P.  u )  =  ( y  +P.  (
w  +P.  u )
)
161, 2, 3, 4, 5, 9, 13, 14, 15ecovass 6975 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  +R  B
)  +R  C )  =  ( A  +R  ( B  +R  C
) ) )
17 dmaddsr 8916 . . 3  |-  dom  +R  =  ( R.  X.  R. )
18 0nsr 8910 . . 3  |-  -.  (/)  e.  R.
1917, 18ndmovass 6194 . 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 1721  (class class class)co 6040   P.cnp 8690    +P. cpp 8692    ~R cer 8697   R.cnr 8698    +R cplr 8702
This theorem is referenced by:  map2psrpr  8941  axaddass  8987  axmulass  8988  axdistr  8989
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-ltpq 8743  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749  df-rq 8750  df-ltnq 8751  df-np 8814  df-plp 8816  df-ltp 8818  df-plpr 8888  df-enr 8890  df-nr 8891  df-plr 8892
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