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Theorem distrsr 8900
Description: Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
distrsr  |-  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) )

Proof of Theorem distrsr
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 8869 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 8884 . . 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  )
3 mulsrpr 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 )
)  +P.  ( y  .P.  ( w  +P.  u
) ) ) ,  ( ( x  .P.  ( w  +P.  u ) )  +P.  ( y  .P.  ( z  +P.  v ) ) )
>. ]  ~R  )
4 mulsrpr 8885 . . 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  )
5 mulsrpr 8885 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( x  .P.  v
)  +P.  ( y  .P.  u ) ) ,  ( ( x  .P.  u )  +P.  (
y  .P.  v )
) >. ]  ~R  )
6 addsrpr 8884 . . 3  |-  ( ( ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P.  /\  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )  /\  ( ( ( x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P.  /\  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
)  ->  ( [ <. ( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >. ]  ~R  +R  [ <. ( ( x  .P.  v
)  +P.  ( y  .P.  u ) ) ,  ( ( x  .P.  u )  +P.  (
y  .P.  v )
) >. ]  ~R  )  =  [ <. ( ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  +P.  ( ( x  .P.  v )  +P.  (
y  .P.  u )
) ) ,  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( (
x  .P.  u )  +P.  ( y  .P.  v
) ) ) >. ]  ~R  )
7 addclpr 8829 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
8 addclpr 8829 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
97, 8anim12i 550 . . . 4  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
109an4s 800 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
11 mulclpr 8831 . . . . . 6  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
12 mulclpr 8831 . . . . . 6  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
13 addclpr 8829 . . . . . 6  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
1411, 12, 13syl2an 464 . . . . 5  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
1514an4s 800 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
16 mulclpr 8831 . . . . . 6  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
17 mulclpr 8831 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
18 addclpr 8829 . . . . . 6  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1916, 17, 18syl2an 464 . . . . 5  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2019an42s 801 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2115, 20jca 519 . . 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. ) )
22 mulclpr 8831 . . . . . 6  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
23 mulclpr 8831 . . . . . 6  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
24 addclpr 8829 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
2522, 23, 24syl2an 464 . . . . 5  |-  ( ( ( x  e.  P.  /\  v  e.  P. )  /\  ( y  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
2625an4s 800 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
27 mulclpr 8831 . . . . . 6  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
28 mulclpr 8831 . . . . . 6  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
29 addclpr 8829 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3027, 28, 29syl2an 464 . . . . 5  |-  ( ( ( x  e.  P.  /\  u  e.  P. )  /\  ( y  e.  P.  /\  v  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3130an42s 801 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3226, 31jca 519 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( x  .P.  v
)  +P.  ( y  .P.  u ) )  e. 
P.  /\  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. ) )
33 distrpr 8839 . . . . 5  |-  ( x  .P.  ( z  +P.  v ) )  =  ( ( x  .P.  z )  +P.  (
x  .P.  v )
)
34 distrpr 8839 . . . . 5  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
3533, 34oveq12i 6033 . . . 4  |-  ( ( x  .P.  ( z  +P.  v ) )  +P.  ( y  .P.  ( w  +P.  u
) ) )  =  ( ( ( x  .P.  z )  +P.  ( x  .P.  v
) )  +P.  (
( y  .P.  w
)  +P.  ( y  .P.  u ) ) )
36 ovex 6046 . . . . 5  |-  ( x  .P.  z )  e. 
_V
37 ovex 6046 . . . . 5  |-  ( x  .P.  v )  e. 
_V
38 ovex 6046 . . . . 5  |-  ( y  .P.  w )  e. 
_V
39 addcompr 8832 . . . . 5  |-  ( f  +P.  g )  =  ( g  +P.  f
)
40 addasspr 8833 . . . . 5  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
41 ovex 6046 . . . . 5  |-  ( y  .P.  u )  e. 
_V
4236, 37, 38, 39, 40, 41caov4 6218 . . . 4  |-  ( ( ( x  .P.  z
)  +P.  ( x  .P.  v ) )  +P.  ( ( y  .P.  w )  +P.  (
y  .P.  u )
) )  =  ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  +P.  ( (
x  .P.  v )  +P.  ( y  .P.  u
) ) )
4335, 42eqtri 2408 . . 3  |-  ( ( x  .P.  ( z  +P.  v ) )  +P.  ( y  .P.  ( w  +P.  u
) ) )  =  ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  +P.  (
( x  .P.  v
)  +P.  ( y  .P.  u ) ) )
44 distrpr 8839 . . . . 5  |-  ( x  .P.  ( w  +P.  u ) )  =  ( ( x  .P.  w )  +P.  (
x  .P.  u )
)
45 distrpr 8839 . . . . 5  |-  ( y  .P.  ( z  +P.  v ) )  =  ( ( y  .P.  z )  +P.  (
y  .P.  v )
)
4644, 45oveq12i 6033 . . . 4  |-  ( ( x  .P.  ( w  +P.  u ) )  +P.  ( y  .P.  ( z  +P.  v
) ) )  =  ( ( ( x  .P.  w )  +P.  ( x  .P.  u
) )  +P.  (
( y  .P.  z
)  +P.  ( y  .P.  v ) ) )
47 ovex 6046 . . . . 5  |-  ( x  .P.  w )  e. 
_V
48 ovex 6046 . . . . 5  |-  ( x  .P.  u )  e. 
_V
49 ovex 6046 . . . . 5  |-  ( y  .P.  z )  e. 
_V
50 ovex 6046 . . . . 5  |-  ( y  .P.  v )  e. 
_V
5147, 48, 49, 39, 40, 50caov4 6218 . . . 4  |-  ( ( ( x  .P.  w
)  +P.  ( x  .P.  u ) )  +P.  ( ( y  .P.  z )  +P.  (
y  .P.  v )
) )  =  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( (
x  .P.  u )  +P.  ( y  .P.  v
) ) )
5246, 51eqtri 2408 . . 3  |-  ( ( x  .P.  ( w  +P.  u ) )  +P.  ( y  .P.  ( z  +P.  v
) ) )  =  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
( x  .P.  u
)  +P.  ( y  .P.  v ) ) )
531, 2, 3, 4, 5, 6, 10, 21, 32, 43, 52ecovdi 6954 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) ) )
54 dmaddsr 8894 . . 3  |-  dom  +R  =  ( R.  X.  R. )
55 0nsr 8888 . . 3  |-  -.  (/)  e.  R.
56 dmmulsr 8895 . . 3  |-  dom  .R  =  ( R.  X.  R. )
5754, 55, 56ndmovdistr 6176 . 2  |-  ( -.  ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C
) ) )
5853, 57pm2.61i 158 1  |-  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6021   P.cnp 8668    +P. cpp 8670    .P. cmp 8671    ~R cer 8675   R.cnr 8676    +R cplr 8680    .R cmr 8681
This theorem is referenced by:  pn0sr  8910  axmulass  8966  axdistr  8967
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-ec 6844  df-qs 6848  df-ni 8683  df-pli 8684  df-mi 8685  df-lti 8686  df-plpq 8719  df-mpq 8720  df-ltpq 8721  df-enq 8722  df-nq 8723  df-erq 8724  df-plq 8725  df-mq 8726  df-1nq 8727  df-rq 8728  df-ltnq 8729  df-np 8792  df-plp 8794  df-mp 8795  df-ltp 8796  df-plpr 8866  df-mpr 8867  df-enr 8868  df-nr 8869  df-plr 8870  df-mr 8871
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