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Theorem crreczi 11431
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 9035 . . . . . . 7  |-  A  e.  CC
32sqcli 11389 . . . . . 6  |-  ( A ^ 2 )  e.  CC
4 ax-icn 8982 . . . . . . . 8  |-  _i  e.  CC
5 crrecz.2 . . . . . . . . 9  |-  B  e.  RR
65recni 9035 . . . . . . . 8  |-  B  e.  CC
74, 6mulcli 9028 . . . . . . 7  |-  ( _i  x.  B )  e.  CC
87sqcli 11389 . . . . . 6  |-  ( ( _i  x.  B ) ^ 2 )  e.  CC
93, 8negsubi 9310 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  -  (
( _i  x.  B
) ^ 2 ) )
104, 6sqmuli 11392 . . . . . . . . 9  |-  ( ( _i  x.  B ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( B ^ 2 ) )
11 i2 11408 . . . . . . . . . 10  |-  ( _i
^ 2 )  = 
-u 1
1211oveq1i 6030 . . . . . . . . 9  |-  ( ( _i ^ 2 )  x.  ( B ^
2 ) )  =  ( -u 1  x.  ( B ^ 2 ) )
13 ax-1cn 8981 . . . . . . . . . 10  |-  1  e.  CC
146sqcli 11389 . . . . . . . . . 10  |-  ( B ^ 2 )  e.  CC
1513, 14mulneg1i 9411 . . . . . . . . 9  |-  ( -u
1  x.  ( B ^ 2 ) )  =  -u ( 1  x.  ( B ^ 2 ) )
1610, 12, 153eqtri 2411 . . . . . . . 8  |-  ( ( _i  x.  B ) ^ 2 )  = 
-u ( 1  x.  ( B ^ 2 ) )
1716negeqi 9231 . . . . . . 7  |-  -u (
( _i  x.  B
) ^ 2 )  =  -u -u ( 1  x.  ( B ^ 2 ) )
1813, 14mulcli 9028 . . . . . . . 8  |-  ( 1  x.  ( B ^
2 ) )  e.  CC
1918negnegi 9302 . . . . . . 7  |-  -u -u (
1  x.  ( B ^ 2 ) )  =  ( 1  x.  ( B ^ 2 ) )
2014mulid2i 9026 . . . . . . 7  |-  ( 1  x.  ( B ^
2 ) )  =  ( B ^ 2 )
2117, 19, 203eqtri 2411 . . . . . 6  |-  -u (
( _i  x.  B
) ^ 2 )  =  ( B ^
2 )
2221oveq2i 6031 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
232, 7subsqi 11419 . . . . 5  |-  ( ( A ^ 2 )  -  ( ( _i  x.  B ) ^
2 ) )  =  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )
249, 22, 233eqtr3ri 2416 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B
) ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
2524oveq1i 6030 . . 3  |-  ( ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
26 neorian 2637 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
27 sumsqeq0 11387 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  0 ) )
281, 5, 27mp2an 654 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  0 )
2928necon3bbii 2581 . . . . 5  |-  ( -.  ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0
)
3026, 29bitri 241 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )
312, 7addcli 9027 . . . . 5  |-  ( A  +  ( _i  x.  B ) )  e.  CC
322, 7subcli 9308 . . . . 5  |-  ( A  -  ( _i  x.  B ) )  e.  CC
333, 14addcli 9027 . . . . 5  |-  ( ( A ^ 2 )  +  ( B ^
2 ) )  e.  CC
3431, 32, 33divasszi 9696 . . . 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 188 . . 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 9637 . . . . 5  |-  ( ( ( ( A ^
2 )  +  ( B ^ 2 ) )  e.  CC  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3733, 36mpan 652 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  =  1 )
3830, 37sylbi 188 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3925, 35, 383eqtr3a 2443 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 )
4032, 33divclzi 9681 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  e.  CC )
4130, 40sylbi 188 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  -  ( _i  x.  B ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  e.  CC )
4231a1i 11 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
43 crne0 9925 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0 ) )
441, 5, 43mp2an 654 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( A  +  ( _i  x.  B ) )  =/=  0 )
4544biimpi 187 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  =/=  0
)
46 divmul 9613 . . . 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 1266 . . 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 1182 . 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 224 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924   _ici 8925    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224    / cdiv 9609   2c2 9981   ^cexp 11309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-seq 11251  df-exp 11310
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