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Theorem mulasssr 9360
Description: Multiplication 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
mulasssr  |-  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) )

Proof of Theorem mulasssr
Dummy variables  f 
g  h  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9330 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 9346 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ,  ( ( x  .P.  w )  +P.  (
y  .P.  z )
) >. ]  ~R  )
3 mulsrpr 9346 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  .R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( z  .P.  v
)  +P.  ( w  .P.  u ) ) ,  ( ( z  .P.  u )  +P.  (
w  .P.  v )
) >. ]  ~R  )
4 mulsrpr 9346 . . 3  |-  ( ( ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P.  /\  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. ( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >. ]  ~R  .R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  .P.  v
)  +P.  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  .P.  u ) ) ,  ( ( ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  .P.  u )  +P.  (
( ( x  .P.  w )  +P.  (
y  .P.  z )
)  .P.  v )
) >. ]  ~R  )
5 mulsrpr 9346 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( ( z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P.  /\  ( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. ( ( z  .P.  v
)  +P.  ( w  .P.  u ) ) ,  ( ( z  .P.  u )  +P.  (
w  .P.  v )
) >. ]  ~R  )  =  [ <. ( ( x  .P.  ( ( z  .P.  v )  +P.  ( w  .P.  u
) ) )  +P.  ( y  .P.  (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) ) ) ,  ( ( x  .P.  ( ( z  .P.  u )  +P.  ( w  .P.  v ) ) )  +P.  ( y  .P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) ) ) >. ]  ~R  )
6 mulclpr 9292 . . . . . 6  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
7 mulclpr 9292 . . . . . 6  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
8 addclpr 9290 . . . . . 6  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
96, 7, 8syl2an 477 . . . . 5  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
109an4s 822 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
11 mulclpr 9292 . . . . . 6  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
12 mulclpr 9292 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
13 addclpr 9290 . . . . . 6  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1411, 12, 13syl2an 477 . . . . 5  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1514an42s 823 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1610, 15jca 532 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  e. 
P.  /\  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. ) )
17 mulclpr 9292 . . . . . 6  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
18 mulclpr 9292 . . . . . 6  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
19 addclpr 9290 . . . . . 6  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
2017, 18, 19syl2an 477 . . . . 5  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
2120an4s 822 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
22 mulclpr 9292 . . . . . 6  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
23 mulclpr 9292 . . . . . 6  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
24 addclpr 9290 . . . . . 6  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
2522, 23, 24syl2an 477 . . . . 5  |-  ( ( ( z  e.  P.  /\  u  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
2625an42s 823 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
2721, 26jca 532 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( z  .P.  v
)  +P.  ( w  .P.  u ) )  e. 
P.  /\  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. ) )
28 vex 3073 . . . 4  |-  x  e. 
_V
29 vex 3073 . . . 4  |-  y  e. 
_V
30 vex 3073 . . . 4  |-  z  e. 
_V
31 mulcompr 9295 . . . 4  |-  ( f  .P.  g )  =  ( g  .P.  f
)
32 distrpr 9300 . . . 4  |-  ( f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
)
33 vex 3073 . . . 4  |-  w  e. 
_V
34 vex 3073 . . . 4  |-  v  e. 
_V
35 mulasspr 9296 . . . 4  |-  ( ( f  .P.  g )  .P.  h )  =  ( f  .P.  (
g  .P.  h )
)
36 vex 3073 . . . 4  |-  u  e. 
_V
37 addcompr 9293 . . . 4  |-  ( f  +P.  g )  =  ( g  +P.  f
)
38 addasspr 9294 . . . 4  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
3928, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38caovlem2 6401 . . 3  |-  ( ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  .P.  v )  +P.  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  .P.  u
) )  =  ( ( x  .P.  (
( z  .P.  v
)  +P.  ( w  .P.  u ) ) )  +P.  ( y  .P.  ( ( z  .P.  u )  +P.  (
w  .P.  v )
) ) )
4028, 29, 30, 31, 32, 33, 36, 35, 34, 37, 38caovlem2 6401 . . 3  |-  ( ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  .P.  u )  +P.  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  .P.  v
) )  =  ( ( x  .P.  (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) )  +P.  ( y  .P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) ) )
411, 2, 3, 4, 5, 16, 27, 39, 40ecovass 7314 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  .R  B
)  .R  C )  =  ( A  .R  ( B  .R  C ) ) )
42 dmmulsr 9356 . . 3  |-  dom  .R  =  ( R.  X.  R. )
43 0nsr 9349 . . 3  |-  -.  (/)  e.  R.
4442, 43ndmovass 6353 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  .R  B )  .R  C
)  =  ( A  .R  ( B  .R  C ) ) )
4541, 44pm2.61i 164 1  |-  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758  (class class class)co 6192   P.cnp 9129    +P. cpp 9131    .P. cmp 9132    ~R cer 9136   R.cnr 9137    .R cmr 9142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-omul 7027  df-er 7203  df-ec 7205  df-qs 7209  df-ni 9144  df-pli 9145  df-mi 9146  df-lti 9147  df-plpq 9180  df-mpq 9181  df-ltpq 9182  df-enq 9183  df-nq 9184  df-erq 9185  df-plq 9186  df-mq 9187  df-1nq 9188  df-rq 9189  df-ltnq 9190  df-np 9253  df-plp 9255  df-mp 9256  df-ltp 9257  df-mpr 9328  df-enr 9329  df-nr 9330  df-mr 9332
This theorem is referenced by:  sqgt0sr  9376  recexsr  9377  axmulass  9427
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