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Theorem mulreim 8232
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 499 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7666 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7590 . . . . 5  |-  _i  e.  CC
43a1i 9 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 500 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7666 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7658 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
8 simprl 501 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
98recnd 7666 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
10 simprr 502 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1110recnd 7666 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
124, 11mulcld 7658 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
132, 7, 9, 12muladdd 8045 . 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 7788 . . . . . 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 8211 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
1615oveq1i 5716 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( D  x.  B ) )  =  ( -u 1  x.  ( D  x.  B
) )
1714, 16syl6eq 2148 . . . . 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 7658 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  e.  CC )
1918mulm1d 8039 . . . . 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 7659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  =  ( B  x.  D ) )
2120negeqd 7828 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  -u ( D  x.  B
)  =  -u ( B  x.  D )
)
2217, 19, 213eqtrd 2136 . . . 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 5722 . . 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 7658 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  A
)  e.  CC )
254, 24mulcld 7658 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( D  x.  A )
)  e.  CC )
269, 6mulcld 7658 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  x.  B
)  e.  CC )
274, 26mulcld 7658 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( C  x.  B )
)  e.  CC )
2825, 27addcomd 7784 . . . 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 7785 . . . . . 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 7659 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  D
)  =  ( D  x.  A ) )
3130oveq2d 5722 . . . . . 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 2132 . . . . 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 7785 . . . . 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 5724 . . . 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 7662 . . . 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 2142 . . 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 5724 . 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 2132 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 103    = wceq 1299    e. wcel 1448  (class class class)co 5706   CCcc 7498   RRcr 7499   1c1 7501   _ici 7502    + caddc 7503    x. cmul 7505   -ucneg 7805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-setind 4390  ax-resscn 7587  ax-1cn 7588  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-sub 7806  df-neg 7807
This theorem is referenced by:  mulext1  8240
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