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Theorem mulsrpr 8885
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 4369 . 2  |-  <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>.  e.  _V
2 opex 4369 . 2  |-  <. (
( a  .P.  g
)  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  e.  _V
3 opex 4369 . 2  |-  <. (
( c  .P.  t
)  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >.  e.  _V
4 enrex 8879 . 2  |-  ~R  e.  _V
5 enrer 8877 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 8868 . 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 6030 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 6030 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2403 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
109an42s 801 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
11 oveq12 6030 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 6030 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2403 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
1413an42s 801 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
15 df-mpr 8867 . 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 6030 . . . . 5  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  .P.  u
)  =  ( a  .P.  g ) )
17 oveq12 6030 . . . . 5  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .P.  f
)  =  ( b  .P.  h ) )
1816, 17oveqan12d 6040 . . . 4  |-  ( ( ( w  =  a  /\  u  =  g )  /\  ( v  =  b  /\  f  =  h ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) )
1918an4s 800 . . 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 6030 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .P.  f
)  =  ( a  .P.  h ) )
21 oveq12 6030 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .P.  u
)  =  ( b  .P.  g ) )
2220, 21oveqan12d 6040 . . . 4  |-  ( ( ( w  =  a  /\  f  =  h )  /\  ( v  =  b  /\  u  =  g ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( a  .P.  h
)  +P.  ( b  .P.  g ) ) )
2322an42s 801 . . 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 3935 . 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 6030 . . . . 5  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  .P.  u
)  =  ( c  .P.  t ) )
26 oveq12 6030 . . . . 5  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .P.  f
)  =  ( d  .P.  s ) )
2725, 26oveqan12d 6040 . . . 4  |-  ( ( ( w  =  c  /\  u  =  t )  /\  ( v  =  d  /\  f  =  s ) )  ->  ( ( w  .P.  u )  +P.  ( v  .P.  f
) )  =  ( ( c  .P.  t
)  +P.  ( d  .P.  s ) ) )
2827an4s 800 . . 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 6030 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .P.  f
)  =  ( c  .P.  s ) )
30 oveq12 6030 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .P.  u
)  =  ( d  .P.  t ) )
3129, 30oveqan12d 6040 . . . 4  |-  ( ( ( w  =  c  /\  f  =  s )  /\  ( v  =  d  /\  u  =  t ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) )
3231an42s 801 . . 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 3935 . 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 6030 . . . . 5  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  .P.  u
)  =  ( A  .P.  C ) )
35 oveq12 6030 . . . . 5  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .P.  f
)  =  ( B  .P.  D ) )
3634, 35oveqan12d 6040 . . . 4  |-  ( ( ( w  =  A  /\  u  =  C )  /\  ( v  =  B  /\  f  =  D ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) )
3736an4s 800 . . 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 6030 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .P.  f
)  =  ( A  .P.  D ) )
39 oveq12 6030 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .P.  u
)  =  ( B  .P.  C ) )
4038, 39oveqan12d 6040 . . . 4  |-  ( ( ( w  =  A  /\  f  =  D )  /\  ( v  =  B  /\  u  =  C ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( A  .P.  D )  +P.  ( B  .P.  C ) ) )
4140an42s 801 . . 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 3935 . 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 8871 . 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 8869 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
45 mulcmpblnr 8883 . 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 6951 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3761  (class class class)co 6021   [cec 6840   P.cnp 8668    +P. cpp 8670    .P. cmp 8671    .pR cmpr 8674    ~R cer 8675   R.cnr 8676    .R cmr 8681
This theorem is referenced by:  mulclsr  8893  mulcomsr  8898  mulasssr  8899  distrsr  8900  m1m1sr  8902  1idsr  8907  00sr  8908  recexsrlem  8912  mulgt0sr  8914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-ec 6844  df-qs 6848  df-ni 8683  df-pli 8684  df-mi 8685  df-lti 8686  df-plpq 8719  df-mpq 8720  df-ltpq 8721  df-enq 8722  df-nq 8723  df-erq 8724  df-plq 8725  df-mq 8726  df-1nq 8727  df-rq 8728  df-ltnq 8729  df-np 8792  df-plp 8794  df-mp 8795  df-ltp 8796  df-mpr 8867  df-enr 8868  df-nr 8869  df-mr 8871
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