MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cjreim Structured version   Unicode version

Theorem cjreim 12753
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 9475 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9444 . . . 4  |-  _i  e.  CC
3 recn 9475 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9469 . . . 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 12734 . . 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 12732 . . 3  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
9 cjmul 12735 . . . . 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 12752 . . . . . 6  |-  ( * `
 _i )  = 
-u _i
1211a1i 11 . . . . 5  |-  ( B  e.  RR  ->  (
* `  _i )  =  -u _i )
13 cjre 12732 . . . . 5  |-  ( B  e.  RR  ->  (
* `  B )  =  B )
1412, 13oveq12d 6210 . . . 4  |-  ( B  e.  RR  ->  (
( * `  _i )  x.  ( * `  B ) )  =  ( -u _i  x.  B ) )
15 mulneg1 9884 . . . . 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 2496 . . 3  |-  ( B  e.  RR  ->  (
* `  ( _i  x.  B ) )  = 
-u ( _i  x.  B ) )
188, 17oveqan12d 6211 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( * `  A )  +  ( * `  ( _i  x.  B ) ) )  =  ( A  +  -u ( _i  x.  B ) ) )
19 negsub 9760 . . 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 2496 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 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   _ici 9387    + caddc 9388    x. cmul 9390    - cmin 9698   -ucneg 9699   *ccj 12689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-2 10483  df-cj 12692  df-re 12693  df-im 12694
This theorem is referenced by:  cjreim2  12754  dipcj  24249  lnophmlem2  25558
  Copyright terms: Public domain W3C validator