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Theorem mulreim 8378
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 518 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
21recnd 7806 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  CC )
3 ax-icn 7727 . . . . 5  |-  _i  e.  CC
43a1i 9 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  _i  e.  CC )
5 simplr 519 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
65recnd 7806 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  CC )
74, 6mulcld 7798 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  B
)  e.  CC )
8 simprl 520 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
98recnd 7806 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
10 simprr 521 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
1110recnd 7806 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
124, 11mulcld 7798 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  D
)  e.  CC )
132, 7, 9, 12muladdd 8190 . 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 7929 . . . . . 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 8357 . . . . . . 7  |-  ( _i  x.  _i )  = 
-u 1
1615oveq1i 5784 . . . . . 6  |-  ( ( _i  x.  _i )  x.  ( D  x.  B ) )  =  ( -u 1  x.  ( D  x.  B
) )
1714, 16syl6eq 2188 . . . . 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 7798 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  e.  CC )
1918mulm1d 8184 . . . . 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 7799 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  B
)  =  ( B  x.  D ) )
2120negeqd 7969 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  -u ( D  x.  B
)  =  -u ( B  x.  D )
)
2217, 19, 213eqtrd 2176 . . . 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 5790 . . 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 7798 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( D  x.  A
)  e.  CC )
254, 24mulcld 7798 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( D  x.  A )
)  e.  CC )
269, 6mulcld 7798 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  x.  B
)  e.  CC )
274, 26mulcld 7798 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( _i  x.  ( C  x.  B )
)  e.  CC )
2825, 27addcomd 7925 . . . 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 7926 . . . . . 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 7799 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  D
)  =  ( D  x.  A ) )
3130oveq2d 5790 . . . . . 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 2172 . . . . 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 7926 . . . . 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 5792 . . . 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 7802 . . . 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 2182 . . 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 5792 . 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 2172 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 1331    e. wcel 1480  (class class class)co 5774   CCcc 7630   RRcr 7631   1c1 7633   _ici 7634    + caddc 7635    x. cmul 7637   -ucneg 7946
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7724  ax-1cn 7725  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7947  df-neg 7948
This theorem is referenced by:  mulext1  8386
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