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Theorem mulcnsr 8758
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 4237 . 2  |-  <. (
( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >.  e.  _V
2 oveq1 5865 . . . . 5  |-  ( w  =  A  ->  (
w  .R  u )  =  ( A  .R  u ) )
3 oveq1 5865 . . . . . 6  |-  ( v  =  B  ->  (
v  .R  f )  =  ( B  .R  f ) )
43oveq2d 5874 . . . . 5  |-  ( v  =  B  ->  ( -1R  .R  ( v  .R  f ) )  =  ( -1R  .R  ( B  .R  f ) ) )
52, 4oveqan12d 5877 . . . 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 5865 . . . . 5  |-  ( v  =  B  ->  (
v  .R  u )  =  ( B  .R  u ) )
7 oveq1 5865 . . . . 5  |-  ( w  =  A  ->  (
w  .R  f )  =  ( A  .R  f ) )
86, 7oveqan12rd 5878 . . . 4  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( v  .R  u )  +R  (
w  .R  f )
)  =  ( ( B  .R  u )  +R  ( A  .R  f ) ) )
95, 8opeq12d 3804 . . 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 5866 . . . . 5  |-  ( u  =  C  ->  ( A  .R  u )  =  ( A  .R  C
) )
11 oveq2 5866 . . . . . 6  |-  ( f  =  D  ->  ( B  .R  f )  =  ( B  .R  D
) )
1211oveq2d 5874 . . . . 5  |-  ( f  =  D  ->  ( -1R  .R  ( B  .R  f ) )  =  ( -1R  .R  ( B  .R  D ) ) )
1310, 12oveqan12d 5877 . . . 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 5866 . . . . 5  |-  ( u  =  C  ->  ( B  .R  u )  =  ( B  .R  C
) )
15 oveq2 5866 . . . . 5  |-  ( f  =  D  ->  ( A  .R  f )  =  ( A  .R  D
) )
1614, 15oveqan12d 5877 . . . 4  |-  ( ( u  =  C  /\  f  =  D )  ->  ( ( B  .R  u )  +R  ( A  .R  f ) )  =  ( ( B  .R  C )  +R  ( A  .R  D
) ) )
1713, 16opeq12d 3804 . . 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 2335 . 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 8749 . . 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 8743 . . . . . . 7  |-  CC  =  ( R.  X.  R. )
2120eleq2i 2347 . . . . . 6  |-  ( x  e.  CC  <->  x  e.  ( R.  X.  R. )
)
2220eleq2i 2347 . . . . . 6  |-  ( y  e.  CC  <->  y  e.  ( R.  X.  R. )
)
2321, 22anbi12i 678 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  <->  ( x  e.  ( R. 
X.  R. )  /\  y  e.  ( R.  X.  R. ) ) )
2423anbi1i 676 . . . 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 5903 . . 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 2303 . 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 5984 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 set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687  (class class class)co 5858   {coprab 5859   R.cnr 8489   -1Rcm1r 8492    +R cplr 8493    .R cmr 8494   CCcc 8735    x. cmul 8742
This theorem is referenced by:  mulresr  8761  mulcnsrec  8766  axmulf  8768  axi2m1  8781  axcnre  8786
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-c 8743  df-mul 8749
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