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Theorem mulsrpr 8714
Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
mulsrpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )

Proof of Theorem mulsrpr
Dummy variables  x  y  z  w  v  u  t  s  f 
g  h  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4253 . 2  |-  <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>.  e.  _V
2 opex 4253 . 2  |-  <. (
( a  .P.  g
)  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  e.  _V
3 opex 4253 . 2  |-  <. (
( c  .P.  t
)  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >.  e.  _V
4 enrex 8708 . 2  |-  ~R  e.  _V
5 enrer 8706 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 8697 . 2  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
7 oveq12 5883 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 5883 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2311 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
109an42s 800 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
11 oveq12 5883 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 5883 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2311 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
1413an42s 800 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
15 df-mpr 8696 . 2  |-  .pR  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ) ) }
16 oveq12 5883 . . . . 5  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  .P.  u
)  =  ( a  .P.  g ) )
17 oveq12 5883 . . . . 5  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .P.  f
)  =  ( b  .P.  h ) )
1816, 17oveqan12d 5893 . . . 4  |-  ( ( ( w  =  a  /\  u  =  g )  /\  ( v  =  b  /\  f  =  h ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) )
1918an4s 799 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) )
20 oveq12 5883 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .P.  f
)  =  ( a  .P.  h ) )
21 oveq12 5883 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .P.  u
)  =  ( b  .P.  g ) )
2220, 21oveqan12d 5893 . . . 4  |-  ( ( ( w  =  a  /\  f  =  h )  /\  ( v  =  b  /\  u  =  g ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( a  .P.  h
)  +P.  ( b  .P.  g ) ) )
2322an42s 800 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( a  .P.  h )  +P.  ( b  .P.  g ) ) )
2419, 23opeq12d 3820 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( a  .P.  g )  +P.  ( b  .P.  h
) ) ,  ( ( a  .P.  h
)  +P.  ( b  .P.  g ) ) >.
)
25 oveq12 5883 . . . . 5  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  .P.  u
)  =  ( c  .P.  t ) )
26 oveq12 5883 . . . . 5  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .P.  f
)  =  ( d  .P.  s ) )
2725, 26oveqan12d 5893 . . . 4  |-  ( ( ( w  =  c  /\  u  =  t )  /\  ( v  =  d  /\  f  =  s ) )  ->  ( ( w  .P.  u )  +P.  ( v  .P.  f
) )  =  ( ( c  .P.  t
)  +P.  ( d  .P.  s ) ) )
2827an4s 799 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( ( w  .P.  u )  +P.  ( v  .P.  f
) )  =  ( ( c  .P.  t
)  +P.  ( d  .P.  s ) ) )
29 oveq12 5883 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .P.  f
)  =  ( c  .P.  s ) )
30 oveq12 5883 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .P.  u
)  =  ( d  .P.  t ) )
3129, 30oveqan12d 5893 . . . 4  |-  ( ( ( w  =  c  /\  f  =  s )  /\  ( v  =  d  /\  u  =  t ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) )
3231an42s 800 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) )
3328, 32opeq12d 3820 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( ( w  .P.  u )  +P.  ( v  .P.  f
) ) ,  ( ( w  .P.  f
)  +P.  ( v  .P.  u ) ) >.  =  <. ( ( c  .P.  t )  +P.  ( d  .P.  s
) ) ,  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) >.
)
34 oveq12 5883 . . . . 5  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  .P.  u
)  =  ( A  .P.  C ) )
35 oveq12 5883 . . . . 5  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .P.  f
)  =  ( B  .P.  D ) )
3634, 35oveqan12d 5893 . . . 4  |-  ( ( ( w  =  A  /\  u  =  C )  /\  ( v  =  B  /\  f  =  D ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) )
3736an4s 799 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) )
38 oveq12 5883 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .P.  f
)  =  ( A  .P.  D ) )
39 oveq12 5883 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .P.  u
)  =  ( B  .P.  C ) )
4038, 39oveqan12d 5893 . . . 4  |-  ( ( ( w  =  A  /\  f  =  D )  /\  ( v  =  B  /\  u  =  C ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( A  .P.  D )  +P.  ( B  .P.  C ) ) )
4140an42s 800 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( A  .P.  D )  +P.  ( B  .P.  C ) ) )
4237, 41opeq12d 3820 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >.
)
43 df-mr 8700 . 2  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  ~R  /\  y  =  [ <. c ,  d >. ]  ~R  )  /\  z  =  [
( <. a ,  b
>.  .pR  <. c ,  d
>. ) ]  ~R  )
) }
44 df-nr 8698 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
45 mulcmpblnr 8712 . 2  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  (
c  e.  P.  /\  d  e.  P. )
)  /\  ( (
g  e.  P.  /\  h  e.  P. )  /\  ( t  e.  P.  /\  s  e.  P. )
) )  ->  (
( ( a  +P.  d )  =  ( b  +P.  c )  /\  ( g  +P.  s )  =  ( h  +P.  t ) )  ->  <. ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  ~R  <. ( ( c  .P.  t )  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >. ) )
461, 2, 3, 4, 5, 6, 10, 14, 15, 24, 33, 42, 43, 44, 45ovec 6784 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656  (class class class)co 5874   [cec 6674   P.cnp 8497    +P. cpp 8499    .P. cmp 8500    .pR cmpr 8503    ~R cer 8504   R.cnr 8505    .R cmr 8510
This theorem is referenced by:  mulclsr  8722  mulcomsr  8727  mulasssr  8728  distrsr  8729  m1m1sr  8731  1idsr  8736  00sr  8737  recexsrlem  8741  mulgt0sr  8743
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
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