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

Theorem crreczi 11242
Description: Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Jeff Hankins, 16-Dec-2013.)
Hypotheses
Ref Expression
crrecz.1  |-  A  e.  RR
crrecz.2  |-  B  e.  RR
Assertion
Ref Expression
crreczi  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( 1  / 
( A  +  ( _i  x.  B ) ) )  =  ( ( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) ) )

Proof of Theorem crreczi
StepHypRef Expression
1 crrecz.1 . . . . . . . 8  |-  A  e.  RR
21recni 8865 . . . . . . 7  |-  A  e.  CC
32sqcli 11200 . . . . . 6  |-  ( A ^ 2 )  e.  CC
4 ax-icn 8812 . . . . . . . 8  |-  _i  e.  CC
5 crrecz.2 . . . . . . . . 9  |-  B  e.  RR
65recni 8865 . . . . . . . 8  |-  B  e.  CC
74, 6mulcli 8858 . . . . . . 7  |-  ( _i  x.  B )  e.  CC
87sqcli 11200 . . . . . 6  |-  ( ( _i  x.  B ) ^ 2 )  e.  CC
93, 8negsubi 9140 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  -  (
( _i  x.  B
) ^ 2 ) )
104, 6sqmuli 11203 . . . . . . . . 9  |-  ( ( _i  x.  B ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( B ^ 2 ) )
11 i2 11219 . . . . . . . . . 10  |-  ( _i
^ 2 )  = 
-u 1
1211oveq1i 5884 . . . . . . . . 9  |-  ( ( _i ^ 2 )  x.  ( B ^
2 ) )  =  ( -u 1  x.  ( B ^ 2 ) )
13 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
146sqcli 11200 . . . . . . . . . 10  |-  ( B ^ 2 )  e.  CC
1513, 14mulneg1i 9241 . . . . . . . . 9  |-  ( -u
1  x.  ( B ^ 2 ) )  =  -u ( 1  x.  ( B ^ 2 ) )
1610, 12, 153eqtri 2320 . . . . . . . 8  |-  ( ( _i  x.  B ) ^ 2 )  = 
-u ( 1  x.  ( B ^ 2 ) )
1716negeqi 9061 . . . . . . 7  |-  -u (
( _i  x.  B
) ^ 2 )  =  -u -u ( 1  x.  ( B ^ 2 ) )
1813, 14mulcli 8858 . . . . . . . 8  |-  ( 1  x.  ( B ^
2 ) )  e.  CC
1918negnegi 9132 . . . . . . 7  |-  -u -u (
1  x.  ( B ^ 2 ) )  =  ( 1  x.  ( B ^ 2 ) )
2014mulid2i 8856 . . . . . . 7  |-  ( 1  x.  ( B ^
2 ) )  =  ( B ^ 2 )
2117, 19, 203eqtri 2320 . . . . . 6  |-  -u (
( _i  x.  B
) ^ 2 )  =  ( B ^
2 )
2221oveq2i 5885 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
232, 7subsqi 11230 . . . . 5  |-  ( ( A ^ 2 )  -  ( ( _i  x.  B ) ^
2 ) )  =  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )
249, 22, 233eqtr3ri 2325 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B
) ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
2524oveq1i 5884 . . 3  |-  ( ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
26 neorian 2546 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
27 sumsqeq0 11198 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  0 ) )
281, 5, 27mp2an 653 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  0 )
2928necon3bbii 2490 . . . . 5  |-  ( -.  ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0
)
3026, 29bitri 240 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )
312, 7addcli 8857 . . . . 5  |-  ( A  +  ( _i  x.  B ) )  e.  CC
322, 7subcli 9138 . . . . 5  |-  ( A  -  ( _i  x.  B ) )  e.  CC
333, 14addcli 8857 . . . . 5  |-  ( ( A ^ 2 )  +  ( B ^
2 ) )  e.  CC
3431, 32, 33divasszi 9526 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( A  +  ( _i  x.  B
) )  x.  (
( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) ) ) )
3530, 34sylbi 187 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B
) ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) ) )
36 divid 9467 . . . . 5  |-  ( ( ( ( A ^
2 )  +  ( B ^ 2 ) )  e.  CC  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3733, 36mpan 651 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  =  1 )
3830, 37sylbi 187 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3925, 35, 383eqtr3a 2352 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 )
4032, 33divclzi 9511 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  e.  CC )
4130, 40sylbi 187 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  -  ( _i  x.  B ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  e.  CC )
4231a1i 10 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
43 crne0 9755 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0 ) )
441, 5, 43mp2an 653 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( A  +  ( _i  x.  B ) )  =/=  0 )
4544biimpi 186 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  =/=  0
)
46 divmul 9443 . . . 4  |-  ( ( 1  e.  CC  /\  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  e.  CC  /\  ( ( A  +  ( _i  x.  B
) )  e.  CC  /\  ( A  +  ( _i  x.  B ) )  =/=  0 ) )  ->  ( (
1  /  ( A  +  ( _i  x.  B ) ) )  =  ( ( A  -  ( _i  x.  B ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 ) )
4713, 46mp3an1 1264 . . 3  |-  ( ( ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  e.  CC  /\  ( ( A  +  ( _i  x.  B
) )  e.  CC  /\  ( A  +  ( _i  x.  B ) )  =/=  0 ) )  ->  ( (
1  /  ( A  +  ( _i  x.  B ) ) )  =  ( ( A  -  ( _i  x.  B ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 ) )
4841, 42, 45, 47syl12anc 1180 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( 1  /  ( A  +  ( _i  x.  B
) ) )  =  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  <->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 ) )
4939, 48mpbird 223 1  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( 1  / 
( A  +  ( _i  x.  B ) ) )  =  ( ( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   ^cexp 11120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
  Copyright terms: Public domain W3C validator