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Theorem distrsr 8729
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 8698 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 8713 . . 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 8714 . . 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 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  )
5 mulsrpr 8714 . . 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 8713 . . 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 8658 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
8 addclpr 8658 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
97, 8anim12i 549 . . . 4  |-  ( ( ( z  e.  P.  /\  v  e.  P. )  /\  ( w  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
109an4s 799 . . 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 8660 . . . . . 6  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
12 mulclpr 8660 . . . . . 6  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
13 addclpr 8658 . . . . . 6  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
1411, 12, 13syl2an 463 . . . . 5  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
1514an4s 799 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
16 mulclpr 8660 . . . . . 6  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
17 mulclpr 8660 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
18 addclpr 8658 . . . . . 6  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1916, 17, 18syl2an 463 . . . . 5  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2019an42s 800 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2115, 20jca 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. ) )
22 mulclpr 8660 . . . . . 6  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
23 mulclpr 8660 . . . . . 6  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
24 addclpr 8658 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
2522, 23, 24syl2an 463 . . . . 5  |-  ( ( ( x  e.  P.  /\  v  e.  P. )  /\  ( y  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
2625an4s 799 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
27 mulclpr 8660 . . . . . 6  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
28 mulclpr 8660 . . . . . 6  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
29 addclpr 8658 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3027, 28, 29syl2an 463 . . . . 5  |-  ( ( ( x  e.  P.  /\  u  e.  P. )  /\  ( y  e.  P.  /\  v  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3130an42s 800 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3226, 31jca 518 . . 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 8668 . . . . 5  |-  ( x  .P.  ( z  +P.  v ) )  =  ( ( x  .P.  z )  +P.  (
x  .P.  v )
)
34 distrpr 8668 . . . . 5  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
3533, 34oveq12i 5886 . . . 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 5899 . . . . 5  |-  ( x  .P.  z )  e. 
_V
37 ovex 5899 . . . . 5  |-  ( x  .P.  v )  e. 
_V
38 ovex 5899 . . . . 5  |-  ( y  .P.  w )  e. 
_V
39 addcompr 8661 . . . . 5  |-  ( f  +P.  g )  =  ( g  +P.  f
)
40 addasspr 8662 . . . . 5  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
41 ovex 5899 . . . . 5  |-  ( y  .P.  u )  e. 
_V
4236, 37, 38, 39, 40, 41caov4 6067 . . . 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 2316 . . 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 8668 . . . . 5  |-  ( x  .P.  ( w  +P.  u ) )  =  ( ( x  .P.  w )  +P.  (
x  .P.  u )
)
45 distrpr 8668 . . . . 5  |-  ( y  .P.  ( z  +P.  v ) )  =  ( ( y  .P.  z )  +P.  (
y  .P.  v )
)
4644, 45oveq12i 5886 . . . 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 5899 . . . . 5  |-  ( x  .P.  w )  e. 
_V
48 ovex 5899 . . . . 5  |-  ( x  .P.  u )  e. 
_V
49 ovex 5899 . . . . 5  |-  ( y  .P.  z )  e. 
_V
50 ovex 5899 . . . . 5  |-  ( y  .P.  v )  e. 
_V
5147, 48, 49, 39, 40, 50caov4 6067 . . . 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 2316 . . 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 6787 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) ) )
54 dmaddsr 8723 . . 3  |-  dom  +R  =  ( R.  X.  R. )
55 0nsr 8717 . . 3  |-  -.  (/)  e.  R.
56 dmmulsr 8724 . . 3  |-  dom  .R  =  ( R.  X.  R. )
5754, 55, 56ndmovdistr 6025 . 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 156 1  |-  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .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 cplr 8509    .R cmr 8510
This theorem is referenced by:  pn0sr  8739  axmulass  8795  axdistr  8796
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-plpr 8695  df-mpr 8696  df-enr 8697  df-nr 8698  df-plr 8699  df-mr 8700
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