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Theorem mulsrpr 8943
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 4419 . 2  |-  <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>.  e.  _V
2 opex 4419 . 2  |-  <. (
( a  .P.  g
)  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  e.  _V
3 opex 4419 . 2  |-  <. (
( c  .P.  t
)  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >.  e.  _V
4 enrex 8937 . 2  |-  ~R  e.  _V
5 enrer 8935 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 8926 . 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 6082 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 6082 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2450 . . 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 6082 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 6082 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2450 . . 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 8925 . 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 6082 . . . . 5  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  .P.  u
)  =  ( a  .P.  g ) )
17 oveq12 6082 . . . . 5  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .P.  f
)  =  ( b  .P.  h ) )
1816, 17oveqan12d 6092 . . . 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 6082 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .P.  f
)  =  ( a  .P.  h ) )
21 oveq12 6082 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .P.  u
)  =  ( b  .P.  g ) )
2220, 21oveqan12d 6092 . . . 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 3984 . 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 6082 . . . . 5  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  .P.  u
)  =  ( c  .P.  t ) )
26 oveq12 6082 . . . . 5  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .P.  f
)  =  ( d  .P.  s ) )
2725, 26oveqan12d 6092 . . . 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 6082 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .P.  f
)  =  ( c  .P.  s ) )
30 oveq12 6082 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .P.  u
)  =  ( d  .P.  t ) )
3129, 30oveqan12d 6092 . . . 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 3984 . 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 6082 . . . . 5  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  .P.  u
)  =  ( A  .P.  C ) )
35 oveq12 6082 . . . . 5  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .P.  f
)  =  ( B  .P.  D ) )
3634, 35oveqan12d 6092 . . . 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 6082 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .P.  f
)  =  ( A  .P.  D ) )
39 oveq12 6082 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .P.  u
)  =  ( B  .P.  C ) )
4038, 39oveqan12d 6092 . . . 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 3984 . 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 8929 . 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 8927 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
45 mulcmpblnr 8941 . 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 7006 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 1652    e. wcel 1725   <.cop 3809  (class class class)co 6073   [cec 6895   P.cnp 8726    +P. cpp 8728    .P. cmp 8729    .pR cmpr 8732    ~R cer 8733   R.cnr 8734    .R cmr 8739
This theorem is referenced by:  mulclsr  8951  mulcomsr  8956  mulasssr  8957  distrsr  8958  m1m1sr  8960  1idsr  8965  00sr  8966  recexsrlem  8970  mulgt0sr  8972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-rq 8786  df-ltnq 8787  df-np 8850  df-plp 8852  df-mp 8853  df-ltp 8854  df-mpr 8925  df-enr 8926  df-nr 8927  df-mr 8929
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