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Theorem mulsrpr 8698
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 4237 . 2  |-  <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>.  e.  _V
2 opex 4237 . 2  |-  <. (
( a  .P.  g
)  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  e.  _V
3 opex 4237 . 2  |-  <. (
( c  .P.  t
)  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >.  e.  _V
4 enrex 8692 . 2  |-  ~R  e.  _V
5 enrer 8690 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 8681 . 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 5867 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 5867 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2298 . . 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 5867 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 5867 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2298 . . 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 8680 . 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 5867 . . . . 5  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  .P.  u
)  =  ( a  .P.  g ) )
17 oveq12 5867 . . . . 5  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .P.  f
)  =  ( b  .P.  h ) )
1816, 17oveqan12d 5877 . . . 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 5867 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .P.  f
)  =  ( a  .P.  h ) )
21 oveq12 5867 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .P.  u
)  =  ( b  .P.  g ) )
2220, 21oveqan12d 5877 . . . 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 3804 . 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 5867 . . . . 5  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  .P.  u
)  =  ( c  .P.  t ) )
26 oveq12 5867 . . . . 5  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .P.  f
)  =  ( d  .P.  s ) )
2725, 26oveqan12d 5877 . . . 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 5867 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .P.  f
)  =  ( c  .P.  s ) )
30 oveq12 5867 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .P.  u
)  =  ( d  .P.  t ) )
3129, 30oveqan12d 5877 . . . 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 3804 . 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 5867 . . . . 5  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  .P.  u
)  =  ( A  .P.  C ) )
35 oveq12 5867 . . . . 5  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .P.  f
)  =  ( B  .P.  D ) )
3634, 35oveqan12d 5877 . . . 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 5867 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .P.  f
)  =  ( A  .P.  D ) )
39 oveq12 5867 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .P.  u
)  =  ( B  .P.  C ) )
4038, 39oveqan12d 5877 . . . 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 3804 . 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 8684 . 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 8682 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
45 mulcmpblnr 8696 . 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 6768 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 1623    e. wcel 1684   <.cop 3643  (class class class)co 5858   [cec 6658   P.cnp 8481    +P. cpp 8483    .P. cmp 8484    .pR cmpr 8487    ~R cer 8488   R.cnr 8489    .R cmr 8494
This theorem is referenced by:  mulclsr  8706  mulcomsr  8711  mulasssr  8712  distrsr  8713  m1m1sr  8715  1idsr  8720  00sr  8721  recexsrlem  8725  mulgt0sr  8727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-mp 8608  df-ltp 8609  df-mpr 8680  df-enr 8681  df-nr 8682  df-mr 8684
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