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Theorem addasssr 8800
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 8772 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 8787 . . 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 8787 . . 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 8787 . . 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 8787 . . 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 8732 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  +P.  z
)  e.  P. )
7 addclpr 8732 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  +P.  w
)  e.  P. )
86, 7anim12i 549 . . . 4  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  z )  e.  P.  /\  ( y  +P.  w )  e. 
P. ) )
98an4s 799 . . 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 8732 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
11 addclpr 8732 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
1210, 11anim12i 549 . . . 4  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
1312an4s 799 . . 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 8736 . . 3  |-  ( ( x  +P.  z )  +P.  v )  =  ( x  +P.  (
z  +P.  v )
)
15 addasspr 8736 . . 3  |-  ( ( y  +P.  w )  +P.  u )  =  ( y  +P.  (
w  +P.  u )
)
161, 2, 3, 4, 5, 9, 13, 14, 15ecovass 6858 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  +R  B
)  +R  C )  =  ( A  +R  ( B  +R  C
) ) )
17 dmaddsr 8797 . . 3  |-  dom  +R  =  ( R.  X.  R. )
18 0nsr 8791 . . 3  |-  -.  (/)  e.  R.
1917, 18ndmovass 6095 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  +R  B )  +R  C
)  =  ( A  +R  ( B  +R  C ) ) )
2016, 19pm2.61i 156 1  |-  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710  (class class class)co 5945   P.cnp 8571    +P. cpp 8573    ~R cer 8578   R.cnr 8579    +R cplr 8583
This theorem is referenced by:  map2psrpr  8822  axaddass  8868  axmulass  8869  axdistr  8870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-omul 6571  df-er 6747  df-ec 6749  df-qs 6753  df-ni 8586  df-pli 8587  df-mi 8588  df-lti 8589  df-plpq 8622  df-mpq 8623  df-ltpq 8624  df-enq 8625  df-nq 8626  df-erq 8627  df-plq 8628  df-mq 8629  df-1nq 8630  df-rq 8631  df-ltnq 8632  df-np 8695  df-plp 8697  df-ltp 8699  df-plpr 8769  df-enr 8771  df-nr 8772  df-plr 8773
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