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Theorem cjreim 12967
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 9580 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9549 . . . 4  |-  _i  e.  CC
3 recn 9580 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9574 . . . 4  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 663 . . 3  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 cjadd 12948 . . 3  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( * `  ( A  +  (
_i  x.  B )
) )  =  ( ( * `  A
)  +  ( * `
 ( _i  x.  B ) ) ) )
71, 5, 6syl2an 477 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( ( * `
 A )  +  ( * `  (
_i  x.  B )
) ) )
8 cjre 12946 . . 3  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
9 cjmul 12949 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( * `  (
_i  x.  B )
)  =  ( ( * `  _i )  x.  ( * `  B ) ) )
102, 3, 9sylancr 663 . . . 4  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  =  ( ( * `  _i )  x.  (
* `  B )
) )
11 cji 12966 . . . . . 6  |-  ( * `
 _i )  = 
-u _i
1211a1i 11 . . . . 5  |-  ( B  e.  RR  ->  (
* `  _i )  =  -u _i )
13 cjre 12946 . . . . 5  |-  ( B  e.  RR  ->  (
* `  B )  =  B )
1412, 13oveq12d 6295 . . . 4  |-  ( B  e.  RR  ->  (
( * `  _i )  x.  ( * `  B ) )  =  ( -u _i  x.  B ) )
15 mulneg1 9994 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( -u _i  x.  B )  =  -u ( _i  x.  B
) )
162, 3, 15sylancr 663 . . . 4  |-  ( B  e.  RR  ->  ( -u _i  x.  B )  =  -u ( _i  x.  B ) )
1710, 14, 163eqtrd 2486 . . 3  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  = 
-u ( _i  x.  B ) )
188, 17oveqan12d 6296 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( * `  A )  +  ( * `  ( _i  x.  B ) ) )  =  ( A  +  -u ( _i  x.  B ) ) )
19 negsub 9867 . . 3  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  -u ( _i  x.  B
) )  =  ( A  -  ( _i  x.  B ) ) )
201, 5, 19syl2an 477 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  -u ( _i  x.  B
) )  =  ( A  -  ( _i  x.  B ) ) )
217, 18, 203eqtrd 2486 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( A  -  ( _i  x.  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   _ici 9492    + caddc 9493    x. cmul 9495    - cmin 9805   -ucneg 9806   *ccj 12903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-2 10595  df-cj 12906  df-re 12907  df-im 12908
This theorem is referenced by:  cjreim2  12968  dipcj  25492  lnophmlem2  26801
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