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Theorem mulclsr 9363
Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulclsr  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  e.  R. )

Proof of Theorem mulclsr
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9339 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6208 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
32eleq1d 2523 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  e.  ( ( P.  X.  P. ) /.  ~R  )  <->  ( A  .R  [ <. z ,  w >. ]  ~R  )  e.  ( ( P.  X.  P. ) /.  ~R  ) ) )
4 oveq2 6209 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
54eleq1d 2523 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  .R  [
<. z ,  w >. ]  ~R  )  e.  ( ( P.  X.  P. ) /.  ~R  )  <->  ( A  .R  B )  e.  ( ( P.  X.  P. ) /.  ~R  ) ) )
6 mulsrpr 9355 . . . 4  |-  ( ( ( 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  )
7 mulclpr 9301 . . . . . . . 8  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
8 mulclpr 9301 . . . . . . . 8  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
9 addclpr 9299 . . . . . . . 8  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
107, 8, 9syl2an 477 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
1110an4s 822 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
12 mulclpr 9301 . . . . . . . 8  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
13 mulclpr 9301 . . . . . . . 8  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
14 addclpr 9299 . . . . . . . 8  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1512, 13, 14syl2an 477 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1615an42s 823 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1711, 16jca 532 . . . . 5  |-  ( ( ( 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. ) )
18 opelxpi 4980 . . . . 5  |-  ( ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P.  /\  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )  -> 
<. ( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >.  e.  ( P.  X.  P. ) )
19 enrex 9349 . . . . . 6  |-  ~R  e.  _V
2019ecelqsi 7267 . . . . 5  |-  ( <.
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >.  e.  ( P.  X.  P. )  ->  [ <. (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ,  ( ( x  .P.  w )  +P.  (
y  .P.  z )
) >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
2117, 18, 203syl 20 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  [ <. (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ,  ( ( x  .P.  w )  +P.  (
y  .P.  z )
) >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
226, 21eqeltrd 2542 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  e.  ( ( P.  X.  P. ) /.  ~R  ) )
231, 3, 5, 222ecoptocl 7302 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  e.  ( ( P.  X.  P. ) /.  ~R  ) )
2423, 1syl6eleqr 2553 1  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  e.  R. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3992    X. cxp 4947  (class class class)co 6201   [cec 7210   /.cqs 7211   P.cnp 9138    +P. cpp 9140    .P. cmp 9141    ~R cer 9145   R.cnr 9146    .R cmr 9151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-omul 7036  df-er 7212  df-ec 7214  df-qs 7218  df-ni 9153  df-pli 9154  df-mi 9155  df-lti 9156  df-plpq 9189  df-mpq 9190  df-ltpq 9191  df-enq 9192  df-nq 9193  df-erq 9194  df-plq 9195  df-mq 9196  df-1nq 9197  df-rq 9198  df-ltnq 9199  df-np 9262  df-plp 9264  df-mp 9265  df-ltp 9266  df-mpr 9337  df-enr 9338  df-nr 9339  df-mr 9341
This theorem is referenced by:  dmmulsr  9365  negexsr  9381  sqgt0sr  9385  recexsr  9386  map2psrpr  9389  mulresr  9418  axmulf  9425  axmulrcl  9433  axmulass  9436  axdistr  9437  axrnegex  9441
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