MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulclsr Unicode version

Theorem mulclsr 8722
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 8698 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5881 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
32eleq1d 2362 . . 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 5882 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
54eleq1d 2362 . . 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 8714 . . . 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 8660 . . . . . . . 8  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
8 mulclpr 8660 . . . . . . . 8  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
9 addclpr 8658 . . . . . . . 8  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
107, 8, 9syl2an 463 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
1110an4s 799 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
12 mulclpr 8660 . . . . . . . 8  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
13 mulclpr 8660 . . . . . . . 8  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
14 addclpr 8658 . . . . . . . 8  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1512, 13, 14syl2an 463 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1615an42s 800 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1711, 16jca 518 . . . . 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 4737 . . . . 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 8708 . . . . . 6  |-  ~R  e.  _V
2019ecelqsi 6731 . . . . 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 18 . . . 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 2370 . . 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 6765 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  e.  ( ( P.  X.  P. ) /.  ~R  ) )
2423, 1syl6eleqr 2387 1  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( A  .R  B
)  e.  R. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703  (class class class)co 5874   [cec 6674   /.cqs 6675   P.cnp 8497    +P. cpp 8499    .P. cmp 8500    ~R cer 8504   R.cnr 8505    .R cmr 8510
This theorem is referenced by:  dmmulsr  8724  negexsr  8740  sqgt0sr  8744  recexsr  8745  map2psrpr  8748  mulresr  8777  axmulf  8784  axmulrcl  8792  axmulass  8795  axdistr  8796  axrnegex  8800
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-mpr 8696  df-enr 8697  df-nr 8698  df-mr 8700
  Copyright terms: Public domain W3C validator