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Theorem mulcnsr 9562
Description: Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcnsr  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( <. A ,  B >.  x.  <. C ,  D >. )  =  <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D ) ) >.
)

Proof of Theorem mulcnsr
Dummy variables  x  y  z  w  v  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4683 . 2  |-  <. (
( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >.  e.  _V
2 oveq1 6310 . . . . 5  |-  ( w  =  A  ->  (
w  .R  u )  =  ( A  .R  u ) )
3 oveq1 6310 . . . . . 6  |-  ( v  =  B  ->  (
v  .R  f )  =  ( B  .R  f ) )
43oveq2d 6319 . . . . 5  |-  ( v  =  B  ->  ( -1R  .R  ( v  .R  f ) )  =  ( -1R  .R  ( B  .R  f ) ) )
52, 4oveqan12d 6322 . . . 4  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) )  =  ( ( A  .R  u )  +R  ( -1R  .R  ( B  .R  f ) ) ) )
6 oveq1 6310 . . . . 5  |-  ( v  =  B  ->  (
v  .R  u )  =  ( B  .R  u ) )
7 oveq1 6310 . . . . 5  |-  ( w  =  A  ->  (
w  .R  f )  =  ( A  .R  f ) )
86, 7oveqan12rd 6323 . . . 4  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( v  .R  u )  +R  (
w  .R  f )
)  =  ( ( B  .R  u )  +R  ( A  .R  f ) ) )
95, 8opeq12d 4193 . . 3  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >.  =  <. ( ( A  .R  u
)  +R  ( -1R 
.R  ( B  .R  f ) ) ) ,  ( ( B  .R  u )  +R  ( A  .R  f
) ) >. )
10 oveq2 6311 . . . . 5  |-  ( u  =  C  ->  ( A  .R  u )  =  ( A  .R  C
) )
11 oveq2 6311 . . . . . 6  |-  ( f  =  D  ->  ( B  .R  f )  =  ( B  .R  D
) )
1211oveq2d 6319 . . . . 5  |-  ( f  =  D  ->  ( -1R  .R  ( B  .R  f ) )  =  ( -1R  .R  ( B  .R  D ) ) )
1310, 12oveqan12d 6322 . . . 4  |-  ( ( u  =  C  /\  f  =  D )  ->  ( ( A  .R  u )  +R  ( -1R  .R  ( B  .R  f ) ) )  =  ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) )
14 oveq2 6311 . . . . 5  |-  ( u  =  C  ->  ( B  .R  u )  =  ( B  .R  C
) )
15 oveq2 6311 . . . . 5  |-  ( f  =  D  ->  ( A  .R  f )  =  ( A  .R  D
) )
1614, 15oveqan12d 6322 . . . 4  |-  ( ( u  =  C  /\  f  =  D )  ->  ( ( B  .R  u )  +R  ( A  .R  f ) )  =  ( ( B  .R  C )  +R  ( A  .R  D
) ) )
1713, 16opeq12d 4193 . . 3  |-  ( ( u  =  C  /\  f  =  D )  -> 
<. ( ( A  .R  u )  +R  ( -1R  .R  ( B  .R  f ) ) ) ,  ( ( B  .R  u )  +R  ( A  .R  f
) ) >.  =  <. ( ( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >. )
189, 17sylan9eq 2484 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >.  =  <. ( ( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >. )
19 df-mul 9553 . . 3  |-  x.  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
20 df-c 9547 . . . . . . 7  |-  CC  =  ( R.  X.  R. )
2120eleq2i 2501 . . . . . 6  |-  ( x  e.  CC  <->  x  e.  ( R.  X.  R. )
)
2220eleq2i 2501 . . . . . 6  |-  ( y  e.  CC  <->  y  e.  ( R.  X.  R. )
)
2321, 22anbi12i 702 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  <->  ( x  e.  ( R. 
X.  R. )  /\  y  e.  ( R.  X.  R. ) ) )
2423anbi1i 700 . . . 4  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
)  <->  ( ( x  e.  ( R.  X.  R. )  /\  y  e.  ( R.  X.  R. ) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) )
2524oprabbii 6358 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( R.  X.  R. )  /\  y  e.  ( R.  X.  R. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
2619, 25eqtri 2452 . 2  |-  x.  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( R.  X.  R. )  /\  y  e.  ( R.  X.  R. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
271, 18, 26ov3 6445 1  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( <. A ,  B >.  x.  <. C ,  D >. )  =  <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438   E.wex 1660    e. wcel 1869   <.cop 4003    X. cxp 4849  (class class class)co 6303   {coprab 6304   R.cnr 9292   -1Rcm1r 9295    +R cplr 9296    .R cmr 9297   CCcc 9539    x. cmul 9546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-c 9547  df-mul 9553
This theorem is referenced by:  mulresr  9565  mulcnsrec  9570  axmulf  9572  axi2m1  9585  axcnre  9590
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