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Theorem distrsr 9464
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 9430 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 9448 . . 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 9449 . . 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 9449 . . 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 9449 . . 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 9448 . . 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 9392 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
8 addclpr 9392 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
97, 8anim12i 566 . . . 4  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
109an4s 824 . . 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 9394 . . . . . 6  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
12 mulclpr 9394 . . . . . 6  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
13 addclpr 9392 . . . . . 6  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
1411, 12, 13syl2an 477 . . . . 5  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
1514an4s 824 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
16 mulclpr 9394 . . . . . 6  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
17 mulclpr 9394 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
18 addclpr 9392 . . . . . 6  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1916, 17, 18syl2an 477 . . . . 5  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2019an42s 825 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2115, 20jca 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. ) )
22 mulclpr 9394 . . . . . 6  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
23 mulclpr 9394 . . . . . 6  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
24 addclpr 9392 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
2522, 23, 24syl2an 477 . . . . 5  |-  ( ( ( x  e.  P.  /\  v  e.  P. )  /\  ( y  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
2625an4s 824 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
27 mulclpr 9394 . . . . . 6  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
28 mulclpr 9394 . . . . . 6  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
29 addclpr 9392 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3027, 28, 29syl2an 477 . . . . 5  |-  ( ( ( x  e.  P.  /\  u  e.  P. )  /\  ( y  e.  P.  /\  v  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3130an42s 825 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3226, 31jca 532 . . 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 9402 . . . . 5  |-  ( x  .P.  ( z  +P.  v ) )  =  ( ( x  .P.  z )  +P.  (
x  .P.  v )
)
34 distrpr 9402 . . . . 5  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
3533, 34oveq12i 6294 . . . 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 6307 . . . . 5  |-  ( x  .P.  z )  e. 
_V
37 ovex 6307 . . . . 5  |-  ( x  .P.  v )  e. 
_V
38 ovex 6307 . . . . 5  |-  ( y  .P.  w )  e. 
_V
39 addcompr 9395 . . . . 5  |-  ( f  +P.  g )  =  ( g  +P.  f
)
40 addasspr 9396 . . . . 5  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
41 ovex 6307 . . . . 5  |-  ( y  .P.  u )  e. 
_V
4236, 37, 38, 39, 40, 41caov4 6488 . . . 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 2496 . . 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 9402 . . . . 5  |-  ( x  .P.  ( w  +P.  u ) )  =  ( ( x  .P.  w )  +P.  (
x  .P.  u )
)
45 distrpr 9402 . . . . 5  |-  ( y  .P.  ( z  +P.  v ) )  =  ( ( y  .P.  z )  +P.  (
y  .P.  v )
)
4644, 45oveq12i 6294 . . . 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 6307 . . . . 5  |-  ( x  .P.  w )  e. 
_V
48 ovex 6307 . . . . 5  |-  ( x  .P.  u )  e. 
_V
49 ovex 6307 . . . . 5  |-  ( y  .P.  z )  e. 
_V
50 ovex 6307 . . . . 5  |-  ( y  .P.  v )  e. 
_V
5147, 48, 49, 39, 40, 50caov4 6488 . . . 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 2496 . . 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 7416 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) ) )
54 dmaddsr 9458 . . 3  |-  dom  +R  =  ( R.  X.  R. )
55 0nsr 9452 . . 3  |-  -.  (/)  e.  R.
56 dmmulsr 9459 . . 3  |-  dom  .R  =  ( R.  X.  R. )
5754, 55, 56ndmovdistr 6446 . 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 164 1  |-  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6282   P.cnp 9233    +P. cpp 9235    .P. cmp 9236    ~R cer 9238   R.cnr 9239    +R cplr 9243    .R cmr 9244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-plp 9357  df-mp 9358  df-ltp 9359  df-enr 9429  df-nr 9430  df-plr 9431  df-mr 9432
This theorem is referenced by:  pn0sr  9474  axmulass  9530  axdistr  9531
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