Theorem crreczi 12989

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

Step	Hyp	Ref	Expression
1		crrecz.1	. . . . . . . 8 |- A e. RR
2	1	recni 10052	. . . . . . 7 |- A e. CC
3	2	sqcli 12944	. . . . . 6 |- ( A ^ 2 ) e. CC
4		ax-icn 9995	. . . . . . . 8 |- _i e. CC
5		crrecz.2	. . . . . . . . 9 |- B e. RR
6	5	recni 10052	. . . . . . . 8 |- B e. CC
7	4, 6	mulcli 10045	. . . . . . 7 |- ( _i x. B ) e. CC
8	7	sqcli 12944	. . . . . 6 |- ( ( _i x. B ) ^ 2 ) e. CC
9	3, 8	negsubi 10359	. . . . 5 |- ( ( A ^ 2 ) + -u ( ( _i x. B ) ^ 2 ) ) = ( ( A ^ 2 ) - ( ( _i x. B ) ^ 2 ) )
10	4, 6	sqmuli 12947	. . . . . . . . 9 |- ( ( _i x. B ) ^ 2 ) = ( ( _i ^ 2 ) x. ( B ^ 2 ) )
11		i2 12965	. . . . . . . . . 10 |- ( _i ^ 2 ) = -u 1
12	11	oveq1i 6660	. . . . . . . . 9 |- ( ( _i ^ 2 ) x. ( B ^ 2 ) ) = ( -u 1 x. ( B ^ 2 ) )
13		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
14	6	sqcli 12944	. . . . . . . . . 10 |- ( B ^ 2 ) e. CC
15	13, 14	mulneg1i 10476	. . . . . . . . 9 |- ( -u 1 x. ( B ^ 2 ) ) = -u ( 1 x. ( B ^ 2 ) )
16	10, 12, 15	3eqtri 2648	. . . . . . . 8 |- ( ( _i x. B ) ^ 2 ) = -u ( 1 x. ( B ^ 2 ) )
17	16	negeqi 10274	. . . . . . 7 |- -u ( ( _i x. B ) ^ 2 ) = -u -u ( 1 x. ( B ^ 2 ) )
18	13, 14	mulcli 10045	. . . . . . . 8 |- ( 1 x. ( B ^ 2 ) ) e. CC
19	18	negnegi 10351	. . . . . . 7 |- -u -u ( 1 x. ( B ^ 2 ) ) = ( 1 x. ( B ^ 2 ) )
20	14	mulid2i 10043	. . . . . . 7 |- ( 1 x. ( B ^ 2 ) ) = ( B ^ 2 )
21	17, 19, 20	3eqtri 2648	. . . . . 6 |- -u ( ( _i x. B ) ^ 2 ) = ( B ^ 2 )
22	21	oveq2i 6661	. . . . 5 |- ( ( A ^ 2 ) + -u ( ( _i x. B ) ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) )
23	2, 7	subsqi 12975	. . . . 5 |- ( ( A ^ 2 ) - ( ( _i x. B ) ^ 2 ) ) = ( ( A + ( _i x. B ) ) x. ( A - ( _i x. B ) ) )
24	9, 22, 23	3eqtr3ri 2653	. . . 4 |- ( ( A + ( _i x. B ) ) x. ( A - ( _i x. B ) ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) )
25	24	oveq1i 6660	. . 3 |- ( ( ( A + ( _i x. B ) ) x. ( A - ( _i x. B ) ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) )
26		neorian 2888	. . . . 5 |- ( ( A =/= 0 \/ B =/= 0 ) <-> -. ( A = 0 /\ B = 0 ) )
27		sumsqeq0 12942	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 ) )
28	1, 5, 27	mp2an 708	. . . . . 6 |- ( ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 )
29	28	necon3bbii 2841	. . . . 5 |- ( -. ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) =/= 0 )
30	26, 29	bitri 264	. . . 4 |- ( ( A =/= 0 \/ B =/= 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) =/= 0 )
31	2, 7	addcli 10044	. . . . 5 |- ( A + ( _i x. B ) ) e. CC
32	2, 7	subcli 10357	. . . . 5 |- ( A - ( _i x. B ) ) e. CC
33	3, 14	addcli 10044	. . . . 5 |- ( ( A ^ 2 ) + ( B ^ 2 ) ) e. CC
34	31, 32, 33	divasszi 10775	. . . 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 ) ) ) ) )
35	30, 34	sylbi 207	. . 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 10714	. . . . 5 |- ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) e. CC /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) =/= 0 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = 1 )
37	33, 36	mpan 706	. . . 4 |- ( ( ( A ^ 2 ) + ( B ^ 2 ) ) =/= 0 -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = 1 )
38	30, 37	sylbi 207	. . 3 |- ( ( A =/= 0 \/ B =/= 0 ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = 1 )
39	25, 35, 38	3eqtr3a 2680	. 2 |- ( ( A =/= 0 \/ B =/= 0 ) -> ( ( A + ( _i x. B ) ) x. ( ( A - ( _i x. B ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) ) = 1 )
40	32, 33	divclzi 10760	. . . 4 |- ( ( ( A ^ 2 ) + ( B ^ 2 ) ) =/= 0 -> ( ( A - ( _i x. B ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) e. CC )
41	30, 40	sylbi 207	. . 3 |- ( ( A =/= 0 \/ B =/= 0 ) -> ( ( A - ( _i x. B ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) e. CC )
42	31	a1i 11	. . 3 |- ( ( A =/= 0 \/ B =/= 0 ) -> ( A + ( _i x. B ) ) e. CC )
43		crne0 11013	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A =/= 0 \/ B =/= 0 ) <-> ( A + ( _i x. B ) ) =/= 0 ) )
44	1, 5, 43	mp2an 708	. . . 4 |- ( ( A =/= 0 \/ B =/= 0 ) <-> ( A + ( _i x. B ) ) =/= 0 )
45	44	biimpi 206	. . 3 |- ( ( A =/= 0 \/ B =/= 0 ) -> ( A + ( _i x. B ) ) =/= 0 )
46		divmul 10688	. . . 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 ) )
47	13, 46	mp3an1 1411	. . 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 ) )
48	41, 42, 45, 47	syl12anc 1324	. 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 ) )
49	39, 48	mpbird 247	1 |- ( ( A =/= 0 \/ B =/= 0 ) -> ( 1 / ( A + ( _i x. B ) ) ) = ( ( A - ( _i x. B ) ) / ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by: (None)