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Theorem mulasssr 8728
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 8698 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 8714 . . 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 8714 . . 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 8714 . . 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 8714 . . 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 8660 . . . . . 6  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
7 mulclpr 8660 . . . . . 6  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
8 addclpr 8658 . . . . . 6  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
96, 7, 8syl2an 463 . . . . 5  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
109an4s 799 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
11 mulclpr 8660 . . . . . 6  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
12 mulclpr 8660 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
13 addclpr 8658 . . . . . 6  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1411, 12, 13syl2an 463 . . . . 5  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1514an42s 800 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1610, 15jca 518 . . 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 8660 . . . . . 6  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
18 mulclpr 8660 . . . . . 6  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
19 addclpr 8658 . . . . . 6  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
2017, 18, 19syl2an 463 . . . . 5  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
2120an4s 799 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
22 mulclpr 8660 . . . . . 6  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
23 mulclpr 8660 . . . . . 6  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
24 addclpr 8658 . . . . . 6  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
2522, 23, 24syl2an 463 . . . . 5  |-  ( ( ( z  e.  P.  /\  u  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
2625an42s 800 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
2721, 26jca 518 . . 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 2804 . . . 4  |-  x  e. 
_V
29 vex 2804 . . . 4  |-  y  e. 
_V
30 vex 2804 . . . 4  |-  z  e. 
_V
31 mulcompr 8663 . . . 4  |-  ( f  .P.  g )  =  ( g  .P.  f
)
32 distrpr 8668 . . . 4  |-  ( f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
)
33 vex 2804 . . . 4  |-  w  e. 
_V
34 vex 2804 . . . 4  |-  v  e. 
_V
35 mulasspr 8664 . . . 4  |-  ( ( f  .P.  g )  .P.  h )  =  ( f  .P.  (
g  .P.  h )
)
36 vex 2804 . . . 4  |-  u  e. 
_V
37 addcompr 8661 . . . 4  |-  ( f  +P.  g )  =  ( g  +P.  f
)
38 addasspr 8662 . . . 4  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
3928, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38caovlem2 6072 . . 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 6072 . . 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 6786 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  .R  B
)  .R  C )  =  ( A  .R  ( B  .R  C ) ) )
42 dmmulsr 8724 . . 3  |-  dom  .R  =  ( R.  X.  R. )
43 0nsr 8717 . . 3  |-  -.  (/)  e.  R.
4442, 43ndmovass 6024 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  .R  B )  .R  C
)  =  ( A  .R  ( B  .R  C ) ) )
4541, 44pm2.61i 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 1632    e. wcel 1696  (class class class)co 5874   P.cnp 8497    +P. cpp 8499    .P. cmp 8500    ~R cer 8504   R.cnr 8505    .R cmr 8510
This theorem is referenced by:  sqgt0sr  8744  recexsr  8745  axmulass  8795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-mp 8624  df-ltp 8625  df-mpr 8696  df-enr 8697  df-nr 8698  df-mr 8700
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