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Theorem mulreim 8591
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 527 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 8016 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7936 . . . . 5  |-  _i  e.  CC
43a1i 9 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 528 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 8016 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 8008 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
8 simprl 529 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
98recnd 8016 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
10 simprr 531 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1110recnd 8016 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
124, 11mulcld 8008 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
132, 7, 9, 12muladdd 8403 . 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 8142 . . . . . 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 8570 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
1615oveq1i 5906 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( D  x.  B ) )  =  ( -u 1  x.  ( D  x.  B
) )
1714, 16eqtrdi 2238 . . . . 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 8008 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  e.  CC )
1918mulm1d 8397 . . . . 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 8009 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  =  ( B  x.  D ) )
2120negeqd 8182 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  -u ( D  x.  B
)  =  -u ( B  x.  D )
)
2217, 19, 213eqtrd 2226 . . . 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 5912 . . 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 8008 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  A
)  e.  CC )
254, 24mulcld 8008 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( D  x.  A )
)  e.  CC )
269, 6mulcld 8008 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  x.  B
)  e.  CC )
274, 26mulcld 8008 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( C  x.  B )
)  e.  CC )
2825, 27addcomd 8138 . . . 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 8139 . . . . . 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 8009 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  D
)  =  ( D  x.  A ) )
3130oveq2d 5912 . . . . . 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 2222 . . . . 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 8139 . . . . 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 5914 . . . 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 8012 . . . 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 2232 . . 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 5914 . 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 2222 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 104    = wceq 1364    e. wcel 2160  (class class class)co 5896   CCcc 7839   RRcr 7840   1c1 7842   _ici 7843    + caddc 7844    x. cmul 7846   -ucneg 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sub 8160  df-neg 8161
This theorem is referenced by:  mulext1  8599
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