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Theorem mulcnsr 9512
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 4711 . 2  |-  <. (
( A  .R  C
)  +R  ( -1R 
.R  ( B  .R  D ) ) ) ,  ( ( B  .R  C )  +R  ( A  .R  D
) ) >.  e.  _V
2 oveq1 6290 . . . . 5  |-  ( w  =  A  ->  (
w  .R  u )  =  ( A  .R  u ) )
3 oveq1 6290 . . . . . 6  |-  ( v  =  B  ->  (
v  .R  f )  =  ( B  .R  f ) )
43oveq2d 6299 . . . . 5  |-  ( v  =  B  ->  ( -1R  .R  ( v  .R  f ) )  =  ( -1R  .R  ( B  .R  f ) ) )
52, 4oveqan12d 6302 . . . 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 6290 . . . . 5  |-  ( v  =  B  ->  (
v  .R  u )  =  ( B  .R  u ) )
7 oveq1 6290 . . . . 5  |-  ( w  =  A  ->  (
w  .R  f )  =  ( A  .R  f ) )
86, 7oveqan12rd 6303 . . . 4  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( v  .R  u )  +R  (
w  .R  f )
)  =  ( ( B  .R  u )  +R  ( A  .R  f ) ) )
95, 8opeq12d 4221 . . 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 6291 . . . . 5  |-  ( u  =  C  ->  ( A  .R  u )  =  ( A  .R  C
) )
11 oveq2 6291 . . . . . 6  |-  ( f  =  D  ->  ( B  .R  f )  =  ( B  .R  D
) )
1211oveq2d 6299 . . . . 5  |-  ( f  =  D  ->  ( -1R  .R  ( B  .R  f ) )  =  ( -1R  .R  ( B  .R  D ) ) )
1310, 12oveqan12d 6302 . . . 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 6291 . . . . 5  |-  ( u  =  C  ->  ( B  .R  u )  =  ( B  .R  C
) )
15 oveq2 6291 . . . . 5  |-  ( f  =  D  ->  ( A  .R  f )  =  ( A  .R  D
) )
1614, 15oveqan12d 6302 . . . 4  |-  ( ( u  =  C  /\  f  =  D )  ->  ( ( B  .R  u )  +R  ( A  .R  f ) )  =  ( ( B  .R  C )  +R  ( A  .R  D
) ) )
1713, 16opeq12d 4221 . . 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 2528 . 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 9503 . . 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 9497 . . . . . . 7  |-  CC  =  ( R.  X.  R. )
2120eleq2i 2545 . . . . . 6  |-  ( x  e.  CC  <->  x  e.  ( R.  X.  R. )
)
2220eleq2i 2545 . . . . . 6  |-  ( y  e.  CC  <->  y  e.  ( R.  X.  R. )
)
2321, 22anbi12i 697 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  <->  ( x  e.  ( R. 
X.  R. )  /\  y  e.  ( R.  X.  R. ) ) )
2423anbi1i 695 . . . 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 6335 . . 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 2496 . 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 6422 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 369    = wceq 1379   E.wex 1596    e. wcel 1767   <.cop 4033    X. cxp 4997  (class class class)co 6283   {coprab 6284   R.cnr 9242   -1Rcm1r 9245    +R cplr 9246    .R cmr 9247   CCcc 9489    x. cmul 9496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-oprab 6287  df-c 9497  df-mul 9503
This theorem is referenced by:  mulresr  9515  mulcnsrec  9520  axmulf  9522  axi2m1  9535  axcnre  9540
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