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Theorem cjreim 12943
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 9571 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9540 . . . 4  |-  _i  e.  CC
3 recn 9571 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9565 . . . 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 12924 . . 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 12922 . . 3  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
9 cjmul 12925 . . . . 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 12942 . . . . . 6  |-  ( * `
 _i )  = 
-u _i
1211a1i 11 . . . . 5  |-  ( B  e.  RR  ->  (
* `  _i )  =  -u _i )
13 cjre 12922 . . . . 5  |-  ( B  e.  RR  ->  (
* `  B )  =  B )
1412, 13oveq12d 6293 . . . 4  |-  ( B  e.  RR  ->  (
( * `  _i )  x.  ( * `  B ) )  =  ( -u _i  x.  B ) )
15 mulneg1 9982 . . . . 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 2505 . . 3  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  = 
-u ( _i  x.  B ) )
188, 17oveqan12d 6294 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( * `  A )  +  ( * `  ( _i  x.  B ) ) )  =  ( A  +  -u ( _i  x.  B ) ) )
19 negsub 9856 . . 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 2505 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 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9794   -ucneg 9795   *ccj 12879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-2 10583  df-cj 12882  df-re 12883  df-im 12884
This theorem is referenced by:  cjreim2  12944  dipcj  25153  lnophmlem2  26462
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