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Theorem mulreim 8523
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 524 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7948 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7869 . . . . 5  |-  _i  e.  CC
43a1i 9 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 525 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7948 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7940 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
8 simprl 526 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
98recnd 7948 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
10 simprr 527 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1110recnd 7948 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
124, 11mulcld 7940 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
132, 7, 9, 12muladdd 8335 . 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 8074 . . . . . 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 8502 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
1615oveq1i 5863 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( D  x.  B ) )  =  ( -u 1  x.  ( D  x.  B
) )
1714, 16eqtrdi 2219 . . . . 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 7940 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  e.  CC )
1918mulm1d 8329 . . . . 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 7941 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  =  ( B  x.  D ) )
2120negeqd 8114 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  -u ( D  x.  B
)  =  -u ( B  x.  D )
)
2217, 19, 213eqtrd 2207 . . . 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 5869 . . 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 7940 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  A
)  e.  CC )
254, 24mulcld 7940 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( D  x.  A )
)  e.  CC )
269, 6mulcld 7940 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  x.  B
)  e.  CC )
274, 26mulcld 7940 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( C  x.  B )
)  e.  CC )
2825, 27addcomd 8070 . . . 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 8071 . . . . . 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 7941 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  D
)  =  ( D  x.  A ) )
3130oveq2d 5869 . . . . . 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 2203 . . . . 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 8071 . . . . 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 5871 . . . 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 7944 . . . 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 2213 . . 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 5871 . 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 2203 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 1348    e. wcel 2141  (class class class)co 5853   CCcc 7772   RRcr 7773   1c1 7775   _ici 7776    + caddc 7777    x. cmul 7779   -ucneg 8091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093
This theorem is referenced by:  mulext1  8531
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