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Theorem crreczi 11226
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 8849 . . . . . . 7  |-  A  e.  CC
32sqcli 11184 . . . . . 6  |-  ( A ^ 2 )  e.  CC
4 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
5 crrecz.2 . . . . . . . . 9  |-  B  e.  RR
65recni 8849 . . . . . . . 8  |-  B  e.  CC
74, 6mulcli 8842 . . . . . . 7  |-  ( _i  x.  B )  e.  CC
87sqcli 11184 . . . . . 6  |-  ( ( _i  x.  B ) ^ 2 )  e.  CC
93, 8negsubi 9124 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  -  (
( _i  x.  B
) ^ 2 ) )
104, 6sqmuli 11187 . . . . . . . . 9  |-  ( ( _i  x.  B ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( B ^ 2 ) )
11 i2 11203 . . . . . . . . . 10  |-  ( _i
^ 2 )  = 
-u 1
1211oveq1i 5868 . . . . . . . . 9  |-  ( ( _i ^ 2 )  x.  ( B ^
2 ) )  =  ( -u 1  x.  ( B ^ 2 ) )
13 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
146sqcli 11184 . . . . . . . . . 10  |-  ( B ^ 2 )  e.  CC
1513, 14mulneg1i 9225 . . . . . . . . 9  |-  ( -u
1  x.  ( B ^ 2 ) )  =  -u ( 1  x.  ( B ^ 2 ) )
1610, 12, 153eqtri 2307 . . . . . . . 8  |-  ( ( _i  x.  B ) ^ 2 )  = 
-u ( 1  x.  ( B ^ 2 ) )
1716negeqi 9045 . . . . . . 7  |-  -u (
( _i  x.  B
) ^ 2 )  =  -u -u ( 1  x.  ( B ^ 2 ) )
1813, 14mulcli 8842 . . . . . . . 8  |-  ( 1  x.  ( B ^
2 ) )  e.  CC
1918negnegi 9116 . . . . . . 7  |-  -u -u (
1  x.  ( B ^ 2 ) )  =  ( 1  x.  ( B ^ 2 ) )
2014mulid2i 8840 . . . . . . 7  |-  ( 1  x.  ( B ^
2 ) )  =  ( B ^ 2 )
2117, 19, 203eqtri 2307 . . . . . 6  |-  -u (
( _i  x.  B
) ^ 2 )  =  ( B ^
2 )
2221oveq2i 5869 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
232, 7subsqi 11214 . . . . 5  |-  ( ( A ^ 2 )  -  ( ( _i  x.  B ) ^
2 ) )  =  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )
249, 22, 233eqtr3ri 2312 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B
) ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
2524oveq1i 5868 . . 3  |-  ( ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
26 neorian 2533 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
27 sumsqeq0 11182 . . . . . . 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 2477 . . . . 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 8841 . . . . 5  |-  ( A  +  ( _i  x.  B ) )  e.  CC
322, 7subcli 9122 . . . . 5  |-  ( A  -  ( _i  x.  B ) )  e.  CC
333, 14addcli 8841 . . . . 5  |-  ( ( A ^ 2 )  +  ( B ^
2 ) )  e.  CC
3431, 32, 33divasszi 9510 . . . 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 9451 . . . . 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 2339 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 )
4032, 33divclzi 9495 . . . 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 9739 . . . . 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 9427 . . . 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 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   ^cexp 11104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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