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Theorem cjreim 11957
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 9072 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9041 . . . 4  |-  _i  e.  CC
3 recn 9072 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9066 . . . 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 11938 . . 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 11936 . . 3  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
9 cjmul 11939 . . . . 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 11956 . . . . . 6  |-  ( * `
 _i )  = 
-u _i
1211a1i 11 . . . . 5  |-  ( B  e.  RR  ->  (
* `  _i )  =  -u _i )
13 cjre 11936 . . . . 5  |-  ( B  e.  RR  ->  (
* `  B )  =  B )
1412, 13oveq12d 6091 . . . 4  |-  ( B  e.  RR  ->  (
( * `  _i )  x.  ( * `  B ) )  =  ( -u _i  x.  B ) )
15 mulneg1 9462 . . . . 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 2471 . . 3  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  = 
-u ( _i  x.  B ) )
188, 17oveqan12d 6092 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( * `  A )  +  ( * `  ( _i  x.  B ) ) )  =  ( A  +  -u ( _i  x.  B ) ) )
19 negsub 9341 . . 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 2471 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   _ici 8984    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284   *ccj 11893
This theorem is referenced by:  cjreim2  11958  dipcj  22205  lnophmlem2  23512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898
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