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Theorem distrsr 8593
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
StepHypRef Expression
1 df-nr 8562 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 8577 . . 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 8578 . . 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 8578 . . 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 8578 . . 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 8577 . . 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 8522 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
8 addclpr 8522 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
97, 8anim12i 551 . . . 4  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
109an4s 802 . . 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 8524 . . . . . 6  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
12 mulclpr 8524 . . . . . 6  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
13 addclpr 8522 . . . . . 6  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
1411, 12, 13syl2an 465 . . . . 5  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
1514an4s 802 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
16 mulclpr 8524 . . . . . 6  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
17 mulclpr 8524 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
18 addclpr 8522 . . . . . 6  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1916, 17, 18syl2an 465 . . . . 5  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2019an42s 803 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2115, 20jca 520 . . 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 8524 . . . . . 6  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
23 mulclpr 8524 . . . . . 6  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
24 addclpr 8522 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
2522, 23, 24syl2an 465 . . . . 5  |-  ( ( ( x  e.  P.  /\  v  e.  P. )  /\  ( y  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
2625an4s 802 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
27 mulclpr 8524 . . . . . 6  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
28 mulclpr 8524 . . . . . 6  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
29 addclpr 8522 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3027, 28, 29syl2an 465 . . . . 5  |-  ( ( ( x  e.  P.  /\  u  e.  P. )  /\  ( y  e.  P.  /\  v  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3130an42s 803 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3226, 31jca 520 . . 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 8532 . . . . 5  |-  ( x  .P.  ( z  +P.  v ) )  =  ( ( x  .P.  z )  +P.  (
x  .P.  v )
)
34 distrpr 8532 . . . . 5  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
3533, 34oveq12i 5722 . . . 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 5735 . . . . 5  |-  ( x  .P.  z )  e. 
_V
37 ovex 5735 . . . . 5  |-  ( x  .P.  v )  e. 
_V
38 ovex 5735 . . . . 5  |-  ( y  .P.  w )  e. 
_V
39 addcompr 8525 . . . . 5  |-  ( f  +P.  g )  =  ( g  +P.  f
)
40 addasspr 8526 . . . . 5  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
41 ovex 5735 . . . . 5  |-  ( y  .P.  u )  e. 
_V
4236, 37, 38, 39, 40, 41caov4 5903 . . . 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 2273 . . 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 8532 . . . . 5  |-  ( x  .P.  ( w  +P.  u ) )  =  ( ( x  .P.  w )  +P.  (
x  .P.  u )
)
45 distrpr 8532 . . . . 5  |-  ( y  .P.  ( z  +P.  v ) )  =  ( ( y  .P.  z )  +P.  (
y  .P.  v )
)
4644, 45oveq12i 5722 . . . 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 5735 . . . . 5  |-  ( x  .P.  w )  e. 
_V
48 ovex 5735 . . . . 5  |-  ( x  .P.  u )  e. 
_V
49 ovex 5735 . . . . 5  |-  ( y  .P.  z )  e. 
_V
50 ovex 5735 . . . . 5  |-  ( y  .P.  v )  e. 
_V
5147, 48, 49, 39, 40, 50caov4 5903 . . . 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 2273 . . 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 6657 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) ) )
54 dmaddsr 8587 . . 3  |-  dom  +R  =  ( R.  X.  R. )
55 0nsr 8581 . . 3  |-  -.  (/)  e.  R.
56 dmmulsr 8588 . . 3  |-  dom  .R  =  ( R.  X.  R. )
5754, 55, 56ndmovdistr 5861 . 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 360    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5710   P.cnp 8361    +P. cpp 8363    .P. cmp 8364    ~R cer 8368   R.cnr 8369    +R cplr 8373    .R cmr 8374
This theorem is referenced by:  pn0sr  8603  axmulass  8659  axdistr  8660
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6546  df-ec 6548  df-qs 6552  df-ni 8376  df-pli 8377  df-mi 8378  df-lti 8379  df-plpq 8412  df-mpq 8413  df-ltpq 8414  df-enq 8415  df-nq 8416  df-erq 8417  df-plq 8418  df-mq 8419  df-1nq 8420  df-rq 8421  df-ltnq 8422  df-np 8485  df-plp 8487  df-mp 8488  df-ltp 8489  df-plpr 8559  df-mpr 8560  df-enr 8561  df-nr 8562  df-plr 8563  df-mr 8564
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