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Theorem crreczi 12099
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 9502 . . . . . . 7  |-  A  e.  CC
32sqcli 12056 . . . . . 6  |-  ( A ^ 2 )  e.  CC
4 ax-icn 9445 . . . . . . . 8  |-  _i  e.  CC
5 crrecz.2 . . . . . . . . 9  |-  B  e.  RR
65recni 9502 . . . . . . . 8  |-  B  e.  CC
74, 6mulcli 9495 . . . . . . 7  |-  ( _i  x.  B )  e.  CC
87sqcli 12056 . . . . . 6  |-  ( ( _i  x.  B ) ^ 2 )  e.  CC
93, 8negsubi 9790 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  -  (
( _i  x.  B
) ^ 2 ) )
104, 6sqmuli 12059 . . . . . . . . 9  |-  ( ( _i  x.  B ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( B ^ 2 ) )
11 i2 12076 . . . . . . . . . 10  |-  ( _i
^ 2 )  = 
-u 1
1211oveq1i 6203 . . . . . . . . 9  |-  ( ( _i ^ 2 )  x.  ( B ^
2 ) )  =  ( -u 1  x.  ( B ^ 2 ) )
13 ax-1cn 9444 . . . . . . . . . 10  |-  1  e.  CC
146sqcli 12056 . . . . . . . . . 10  |-  ( B ^ 2 )  e.  CC
1513, 14mulneg1i 9894 . . . . . . . . 9  |-  ( -u
1  x.  ( B ^ 2 ) )  =  -u ( 1  x.  ( B ^ 2 ) )
1610, 12, 153eqtri 2484 . . . . . . . 8  |-  ( ( _i  x.  B ) ^ 2 )  = 
-u ( 1  x.  ( B ^ 2 ) )
1716negeqi 9707 . . . . . . 7  |-  -u (
( _i  x.  B
) ^ 2 )  =  -u -u ( 1  x.  ( B ^ 2 ) )
1813, 14mulcli 9495 . . . . . . . 8  |-  ( 1  x.  ( B ^
2 ) )  e.  CC
1918negnegi 9782 . . . . . . 7  |-  -u -u (
1  x.  ( B ^ 2 ) )  =  ( 1  x.  ( B ^ 2 ) )
2014mulid2i 9493 . . . . . . 7  |-  ( 1  x.  ( B ^
2 ) )  =  ( B ^ 2 )
2117, 19, 203eqtri 2484 . . . . . 6  |-  -u (
( _i  x.  B
) ^ 2 )  =  ( B ^
2 )
2221oveq2i 6204 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
232, 7subsqi 12087 . . . . 5  |-  ( ( A ^ 2 )  -  ( ( _i  x.  B ) ^
2 ) )  =  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )
249, 22, 233eqtr3ri 2489 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B
) ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
2524oveq1i 6203 . . 3  |-  ( ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
26 neorian 2775 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
27 sumsqeq0 12054 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  0 ) )
281, 5, 27mp2an 672 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  0 )
2928necon3bbii 2709 . . . . 5  |-  ( -.  ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0
)
3026, 29bitri 249 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )
312, 7addcli 9494 . . . . 5  |-  ( A  +  ( _i  x.  B ) )  e.  CC
322, 7subcli 9788 . . . . 5  |-  ( A  -  ( _i  x.  B ) )  e.  CC
333, 14addcli 9494 . . . . 5  |-  ( ( A ^ 2 )  +  ( B ^
2 ) )  e.  CC
3431, 32, 33divasszi 10185 . . . 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 195 . . 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 10125 . . . . 5  |-  ( ( ( ( A ^
2 )  +  ( B ^ 2 ) )  e.  CC  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3733, 36mpan 670 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  =  1 )
3830, 37sylbi 195 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3925, 35, 383eqtr3a 2516 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 )
4032, 33divclzi 10170 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  e.  CC )
4130, 40sylbi 195 . . 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 10419 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0 ) )
441, 5, 43mp2an 672 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( A  +  ( _i  x.  B ) )  =/=  0 )
4544biimpi 194 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  =/=  0
)
46 divmul 10101 . . . 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 1302 . . 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 1217 . 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 232 1  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( 1  / 
( A  +  ( _i  x.  B ) ) )  =  ( ( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387   _ici 9388    + caddc 9389    x. cmul 9391    - cmin 9699   -ucneg 9700    / cdiv 10097   2c2 10475   ^cexp 11975
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 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
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 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-seq 11917  df-exp 11976
This theorem is referenced by: (None)
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