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Theorem crreczi 11104
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 8729 . . . . . . 7  |-  A  e.  CC
32sqcli 11062 . . . . . 6  |-  ( A ^ 2 )  e.  CC
4 ax-icn 8676 . . . . . . . 8  |-  _i  e.  CC
5 crrecz.2 . . . . . . . . 9  |-  B  e.  RR
65recni 8729 . . . . . . . 8  |-  B  e.  CC
74, 6mulcli 8722 . . . . . . 7  |-  ( _i  x.  B )  e.  CC
87sqcli 11062 . . . . . 6  |-  ( ( _i  x.  B ) ^ 2 )  e.  CC
93, 8negsubi 9004 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  -  (
( _i  x.  B
) ^ 2 ) )
104, 6sqmuli 11065 . . . . . . . . 9  |-  ( ( _i  x.  B ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( B ^ 2 ) )
11 i2 11081 . . . . . . . . . 10  |-  ( _i
^ 2 )  = 
-u 1
1211oveq1i 5720 . . . . . . . . 9  |-  ( ( _i ^ 2 )  x.  ( B ^
2 ) )  =  ( -u 1  x.  ( B ^ 2 ) )
13 ax-1cn 8675 . . . . . . . . . 10  |-  1  e.  CC
146sqcli 11062 . . . . . . . . . 10  |-  ( B ^ 2 )  e.  CC
1513, 14mulneg1i 9105 . . . . . . . . 9  |-  ( -u
1  x.  ( B ^ 2 ) )  =  -u ( 1  x.  ( B ^ 2 ) )
1610, 12, 153eqtri 2277 . . . . . . . 8  |-  ( ( _i  x.  B ) ^ 2 )  = 
-u ( 1  x.  ( B ^ 2 ) )
1716negeqi 8925 . . . . . . 7  |-  -u (
( _i  x.  B
) ^ 2 )  =  -u -u ( 1  x.  ( B ^ 2 ) )
1813, 14mulcli 8722 . . . . . . . 8  |-  ( 1  x.  ( B ^
2 ) )  e.  CC
1918negnegi 8996 . . . . . . 7  |-  -u -u (
1  x.  ( B ^ 2 ) )  =  ( 1  x.  ( B ^ 2 ) )
2014mulid2i 8720 . . . . . . 7  |-  ( 1  x.  ( B ^
2 ) )  =  ( B ^ 2 )
2117, 19, 203eqtri 2277 . . . . . 6  |-  -u (
( _i  x.  B
) ^ 2 )  =  ( B ^
2 )
2221oveq2i 5721 . . . . 5  |-  ( ( A ^ 2 )  +  -u ( ( _i  x.  B ) ^
2 ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
232, 7subsqi 11092 . . . . 5  |-  ( ( A ^ 2 )  -  ( ( _i  x.  B ) ^
2 ) )  =  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )
249, 22, 233eqtr3ri 2282 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B
) ) )  =  ( ( A ^
2 )  +  ( B ^ 2 ) )
2524oveq1i 5720 . . 3  |-  ( ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  /  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
26 neorian 2499 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
27 sumsqeq0 11060 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  0 ) )
281, 5, 27mp2an 656 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =  0 )
2928necon3bbii 2443 . . . . 5  |-  ( -.  ( A  =  0  /\  B  =  0 )  <->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0
)
3026, 29bitri 242 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )
312, 7addcli 8721 . . . . 5  |-  ( A  +  ( _i  x.  B ) )  e.  CC
322, 7subcli 9002 . . . . 5  |-  ( A  -  ( _i  x.  B ) )  e.  CC
333, 14addcli 8721 . . . . 5  |-  ( ( A ^ 2 )  +  ( B ^
2 ) )  e.  CC
3431, 32, 33divasszi 9390 . . . 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 189 . . 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 9331 . . . . 5  |-  ( ( ( ( A ^
2 )  +  ( B ^ 2 ) )  e.  CC  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3733, 36mpan 654 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  =  1 )
3830, 37sylbi 189 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( ( A ^ 2 )  +  ( B ^
2 ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  1 )
3925, 35, 383eqtr3a 2309 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( ( A  -  ( _i  x.  B
) )  /  (
( A ^ 2 )  +  ( B ^ 2 ) ) ) )  =  1 )
4032, 33divclzi 9375 . . . 4  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =/=  0  ->  (
( A  -  (
_i  x.  B )
)  /  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  e.  CC )
4130, 40sylbi 189 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( ( A  -  ( _i  x.  B ) )  / 
( ( A ^
2 )  +  ( B ^ 2 ) ) )  e.  CC )
4231a1i 12 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
43 crne0 9619 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0 ) )
441, 5, 43mp2an 656 . . . 4  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <-> 
( A  +  ( _i  x.  B ) )  =/=  0 )
4544biimpi 188 . . 3  |-  ( ( A  =/=  0  \/  B  =/=  0 )  ->  ( A  +  ( _i  x.  B
) )  =/=  0
)
46 divmul 9307 . . . 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 1269 . . 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 1185 . 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 225 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618   _ici 8619    + caddc 8620    x. cmul 8622    - cmin 8917   -ucneg 8918    / cdiv 9303   2c2 9675   ^cexp 10982
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-n0 9845  df-z 9904  df-uz 10110  df-seq 10925  df-exp 10983
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