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Theorem mulreim 8015
Description: Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
Assertion
Ref Expression
mulreim  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  x.  ( C  +  ( _i  x.  D ) ) )  =  ( ( ( A  x.  C )  +  -u ( B  x.  D ) )  +  ( _i  x.  (
( C  x.  B
)  +  ( D  x.  A ) ) ) ) )

Proof of Theorem mulreim
StepHypRef Expression
1 simpll 496 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7453 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7377 . . . . 5  |-  _i  e.  CC
43a1i 9 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 497 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7453 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7445 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
8 simprl 498 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
98recnd 7453 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
10 simprr 499 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1110recnd 7453 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
124, 11mulcld 7445 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
132, 7, 9, 12muladdd 7831 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  x.  ( C  +  ( _i  x.  D ) ) )  =  ( ( ( A  x.  C )  +  ( ( _i  x.  D )  x.  ( _i  x.  B
) ) )  +  ( ( A  x.  ( _i  x.  D
) )  +  ( C  x.  ( _i  x.  B ) ) ) ) )
144, 11, 4, 6mul4d 7574 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( _i  x.  D )  x.  (
_i  x.  B )
)  =  ( ( _i  x.  _i )  x.  ( D  x.  B ) ) )
15 ixi 7994 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
1615oveq1i 5617 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( D  x.  B ) )  =  ( -u 1  x.  ( D  x.  B
) )
1714, 16syl6eq 2133 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( _i  x.  D )  x.  (
_i  x.  B )
)  =  ( -u
1  x.  ( D  x.  B ) ) )
1811, 6mulcld 7445 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  e.  CC )
1918mulm1d 7825 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( -u 1  x.  ( D  x.  B )
)  =  -u ( D  x.  B )
)
2011, 6mulcomd 7446 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  =  ( B  x.  D ) )
2120negeqd 7614 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  -u ( D  x.  B
)  =  -u ( B  x.  D )
)
2217, 19, 213eqtrd 2121 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( _i  x.  D )  x.  (
_i  x.  B )
)  =  -u ( B  x.  D )
)
2322oveq2d 5623 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  x.  C )  +  ( ( _i  x.  D
)  x.  ( _i  x.  B ) ) )  =  ( ( A  x.  C )  +  -u ( B  x.  D ) ) )
2411, 2mulcld 7445 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  A
)  e.  CC )
254, 24mulcld 7445 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( D  x.  A )
)  e.  CC )
269, 6mulcld 7445 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  x.  B
)  e.  CC )
274, 26mulcld 7445 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( C  x.  B )
)  e.  CC )
2825, 27addcomd 7570 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( _i  x.  ( D  x.  A
) )  +  ( _i  x.  ( C  x.  B ) ) )  =  ( ( _i  x.  ( C  x.  B ) )  +  ( _i  x.  ( D  x.  A
) ) ) )
292, 4, 11mul12d 7571 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  (
_i  x.  D )
)  =  ( _i  x.  ( A  x.  D ) ) )
302, 11mulcomd 7446 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  D
)  =  ( D  x.  A ) )
3130oveq2d 5623 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( A  x.  D )
)  =  ( _i  x.  ( D  x.  A ) ) )
3229, 31eqtrd 2117 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  (
_i  x.  D )
)  =  ( _i  x.  ( D  x.  A ) ) )
339, 4, 6mul12d 7571 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  x.  (
_i  x.  B )
)  =  ( _i  x.  ( C  x.  B ) ) )
3432, 33oveq12d 5625 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  x.  ( _i  x.  D
) )  +  ( C  x.  ( _i  x.  B ) ) )  =  ( ( _i  x.  ( D  x.  A ) )  +  ( _i  x.  ( C  x.  B
) ) ) )
354, 26, 24adddid 7449 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  (
( C  x.  B
)  +  ( D  x.  A ) ) )  =  ( ( _i  x.  ( C  x.  B ) )  +  ( _i  x.  ( D  x.  A
) ) ) )
3628, 34, 353eqtr4d 2127 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  x.  ( _i  x.  D
) )  +  ( C  x.  ( _i  x.  B ) ) )  =  ( _i  x.  ( ( C  x.  B )  +  ( D  x.  A
) ) ) )
3723, 36oveq12d 5625 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( ( A  x.  C )  +  ( ( _i  x.  D )  x.  (
_i  x.  B )
) )  +  ( ( A  x.  (
_i  x.  D )
)  +  ( C  x.  ( _i  x.  B ) ) ) )  =  ( ( ( A  x.  C
)  +  -u ( B  x.  D )
)  +  ( _i  x.  ( ( C  x.  B )  +  ( D  x.  A
) ) ) ) )
3813, 37eqtrd 2117 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  x.  ( C  +  ( _i  x.  D ) ) )  =  ( ( ( A  x.  C )  +  -u ( B  x.  D ) )  +  ( _i  x.  (
( C  x.  B
)  +  ( D  x.  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436  (class class class)co 5607   CCcc 7285   RRcr 7286   1c1 7288   _ici 7289    + caddc 7290    x. cmul 7292   -ucneg 7591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009  ax-setind 4325  ax-resscn 7374  ax-1cn 7375  ax-icn 7377  ax-addcl 7378  ax-addrcl 7379  ax-mulcl 7380  ax-addcom 7382  ax-mulcom 7383  ax-addass 7384  ax-mulass 7385  ax-distr 7386  ax-i2m1 7387  ax-1rid 7389  ax-0id 7390  ax-rnegex 7391  ax-cnre 7393
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4093  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-iota 4943  df-fun 4980  df-fv 4986  df-riota 5563  df-ov 5610  df-oprab 5611  df-mpt2 5612  df-sub 7592  df-neg 7593
This theorem is referenced by:  mulext1  8023
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