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Theorem cjreim 11894
Description: The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
Assertion
Ref Expression
cjreim  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( A  -  ( _i  x.  B
) ) )

Proof of Theorem cjreim
StepHypRef Expression
1 recn 9015 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 8984 . . . 4  |-  _i  e.  CC
3 recn 9015 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9009 . . . 4  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 645 . . 3  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 cjadd 11875 . . 3  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( * `  ( A  +  (
_i  x.  B )
) )  =  ( ( * `  A
)  +  ( * `
 ( _i  x.  B ) ) ) )
71, 5, 6syl2an 464 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( ( * `
 A )  +  ( * `  (
_i  x.  B )
) ) )
8 cjre 11873 . . 3  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
9 cjmul 11876 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( * `  (
_i  x.  B )
)  =  ( ( * `  _i )  x.  ( * `  B ) ) )
102, 3, 9sylancr 645 . . . 4  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  =  ( ( * `  _i )  x.  (
* `  B )
) )
11 cji 11893 . . . . . 6  |-  ( * `
 _i )  = 
-u _i
1211a1i 11 . . . . 5  |-  ( B  e.  RR  ->  (
* `  _i )  =  -u _i )
13 cjre 11873 . . . . 5  |-  ( B  e.  RR  ->  (
* `  B )  =  B )
1412, 13oveq12d 6040 . . . 4  |-  ( B  e.  RR  ->  (
( * `  _i )  x.  ( * `  B ) )  =  ( -u _i  x.  B ) )
15 mulneg1 9404 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( -u _i  x.  B )  =  -u ( _i  x.  B
) )
162, 3, 15sylancr 645 . . . 4  |-  ( B  e.  RR  ->  ( -u _i  x.  B )  =  -u ( _i  x.  B ) )
1710, 14, 163eqtrd 2425 . . 3  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  = 
-u ( _i  x.  B ) )
188, 17oveqan12d 6041 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( * `  A )  +  ( * `  ( _i  x.  B ) ) )  =  ( A  +  -u ( _i  x.  B ) ) )
19 negsub 9283 . . 3  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  -u ( _i  x.  B
) )  =  ( A  -  ( _i  x.  B ) ) )
201, 5, 19syl2an 464 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  -u ( _i  x.  B
) )  =  ( A  -  ( _i  x.  B ) ) )
217, 18, 203eqtrd 2425 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( A  -  ( _i  x.  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   _ici 8927    + caddc 8928    x. cmul 8930    - cmin 9225   -ucneg 9226   *ccj 11830
This theorem is referenced by:  cjreim2  11895  dipcj  22063  lnophmlem2  23370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-2 9992  df-cj 11833  df-re 11834  df-im 11835
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