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Theorem mulreim 7368
Description: Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
Assertion
Ref Expression
mulreim  RR  RR  C  RR  D  RR  +  _i  x.  x.  C  +  _i  x.  D  x.  C  +  -u  x.  D  +  _i  x.  C  x.  +  D  x.

Proof of Theorem mulreim
StepHypRef Expression
1 simpll 481 . . . 4  RR  RR  C  RR  D  RR  RR
21recnd 6831 . . 3  RR  RR  C  RR  D  RR  CC
3 ax-icn 6758 . . . . 5  _i  CC
43a1i 9 . . . 4  RR  RR  C  RR  D  RR  _i  CC
5 simplr 482 . . . . 5  RR  RR  C  RR  D  RR  RR
65recnd 6831 . . . 4  RR  RR  C  RR  D  RR  CC
74, 6mulcld 6825 . . 3  RR  RR  C  RR  D  RR  _i  x.  CC
8 simprl 483 . . . 4  RR  RR  C  RR  D  RR  C  RR
98recnd 6831 . . 3  RR  RR  C  RR  D  RR  C  CC
10 simprr 484 . . . . 5  RR  RR  C  RR  D  RR  D  RR
1110recnd 6831 . . . 4  RR  RR  C  RR  D  RR  D  CC
124, 11mulcld 6825 . . 3  RR  RR  C  RR  D  RR  _i  x.  D  CC
132, 7, 9, 12muladdd 7189 . 2  RR  RR  C  RR  D  RR  +  _i  x.  x.  C  +  _i  x.  D  x.  C  +  _i  x.  D  x.  _i  x.  +  x.  _i  x.  D  +  C  x.  _i  x.
144, 11, 4, 6mul4d 6945 . . . . . 6  RR  RR  C  RR  D  RR  _i  x.  D  x.  _i  x.  _i  x.  _i  x.  D  x.
15 ixi 7347 . . . . . . 7  _i  x.  _i  -u 1
1615oveq1i 5465 . . . . . 6  _i  x.  _i  x.  D  x. 
-u 1  x.  D  x.
1714, 16syl6eq 2085 . . . . 5  RR  RR  C  RR  D  RR  _i  x.  D  x.  _i  x. 
-u 1  x.  D  x.
1811, 6mulcld 6825 . . . . . 6  RR  RR  C  RR  D  RR  D  x.  CC
1918mulm1d 7183 . . . . 5  RR  RR  C  RR  D  RR  -u
1  x.  D  x.  -u D  x.
2011, 6mulcomd 6826 . . . . . 6  RR  RR  C  RR  D  RR  D  x.  x.  D
2120negeqd 6983 . . . . 5  RR  RR  C  RR  D  RR  -u D  x.  -u  x.  D
2217, 19, 213eqtrd 2073 . . . 4  RR  RR  C  RR  D  RR  _i  x.  D  x.  _i  x.  -u  x.  D
2322oveq2d 5471 . . 3  RR  RR  C  RR  D  RR  x.  C  +  _i  x.  D  x.  _i  x.  x.  C  +  -u  x.  D
2411, 2mulcld 6825 . . . . . 6  RR  RR  C  RR  D  RR  D  x.  CC
254, 24mulcld 6825 . . . . 5  RR  RR  C  RR  D  RR  _i  x.  D  x.  CC
269, 6mulcld 6825 . . . . . 6  RR  RR  C  RR  D  RR  C  x.  CC
274, 26mulcld 6825 . . . . 5  RR  RR  C  RR  D  RR  _i  x.  C  x.  CC
2825, 27addcomd 6941 . . . 4  RR  RR  C  RR  D  RR  _i  x.  D  x.  +  _i  x.  C  x.  _i  x.  C  x.  +  _i  x.  D  x.
292, 4, 11mul12d 6942 . . . . . 6  RR  RR  C  RR  D  RR  x.  _i  x.  D  _i  x.  x.  D
302, 11mulcomd 6826 . . . . . . 7  RR  RR  C  RR  D  RR  x.  D  D  x.
3130oveq2d 5471 . . . . . 6  RR  RR  C  RR  D  RR  _i  x.  x.  D  _i  x.  D  x.
3229, 31eqtrd 2069 . . . . 5  RR  RR  C  RR  D  RR  x.  _i  x.  D  _i  x.  D  x.
339, 4, 6mul12d 6942 . . . . 5  RR  RR  C  RR  D  RR  C  x.  _i  x.  _i  x.  C  x.
3432, 33oveq12d 5473 . . . 4  RR  RR  C  RR  D  RR  x.  _i  x.  D  +  C  x.  _i  x.  _i  x.  D  x.  +  _i  x.  C  x.
354, 26, 24adddid 6829 . . . 4  RR  RR  C  RR  D  RR  _i  x.  C  x.  +  D  x.  _i  x.  C  x.  +  _i  x.  D  x.
3628, 34, 353eqtr4d 2079 . . 3  RR  RR  C  RR  D  RR  x.  _i  x.  D  +  C  x.  _i  x.  _i  x.  C  x.  +  D  x.
3723, 36oveq12d 5473 . 2  RR  RR  C  RR  D  RR  x.  C  +  _i  x.  D  x.  _i  x.  +  x.  _i  x.  D  +  C  x.  _i  x.  x.  C  +  -u  x.  D  +  _i  x.  C  x.  +  D  x.
3813, 37eqtrd 2069 1  RR  RR  C  RR  D  RR  +  _i  x.  x.  C  +  _i  x.  D  x.  C  +  -u  x.  D  +  _i  x.  C  x.  +  D  x.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  (class class class)co 5455   CCcc 6689   RRcr 6690   1c1 6692   _ici 6693    + caddc 6694    x. cmul 6696   -ucneg 6960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220  ax-resscn 6755  ax-1cn 6756  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-addcom 6763  ax-mulcom 6764  ax-addass 6765  ax-mulass 6766  ax-distr 6767  ax-i2m1 6768  ax-1rid 6770  ax-0id 6771  ax-rnegex 6772  ax-cnre 6774
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-sub 6961  df-neg 6962
This theorem is referenced by:  mulext1  7376
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