Theorem crreczi 7991

Description: Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361.

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	. . . . . . 7 |- A e. RR
2		crrecz.2	. . . . . . . 8 |- B e. RR
3	2	renegcli 6576	. . . . . . 7 |- -uB e. RR
4	1, 3	crne0i 7989	. . . . . 6 |- ((A =/= 0 \/ -uB =/= 0) <-> (A + (_i x. -uB)) =/= 0)
5	2	recni 6467	. . . . . . . 8 |- B e. CC
6	5	negne0bi 6985	. . . . . . 7 |- (B =/= 0 <-> -uB =/= 0)
7	6	orbi2i 275	. . . . . 6 |- ((A =/= 0 \/ B =/= 0) <-> (A =/= 0 \/ -uB =/= 0))
8		axicn 6423	. . . . . . . . . 10 |- _i e. CC
9	8, 5	mulneg2i 6609	. . . . . . . . 9 |- (_i x. -uB) = -u(_i x. B)
10	9	opreq2i 4893	. . . . . . . 8 |- (A + (_i x. -uB)) = (A + -u(_i x. B))
11	1	recni 6467	. . . . . . . . 9 |- A e. CC
12	8, 5	mulcli 6474	. . . . . . . . 9 |- (_i x. B) e. CC
13	11, 12	negsubi 6538	. . . . . . . 8 |- (A + -u(_i x. B)) = (A - (_i x. B))
14	10, 13	eqtr2i 1909	. . . . . . 7 |- (A - (_i x. B)) = (A + (_i x. -uB))
15	14	neeq1i 2026	. . . . . 6 |- ((A - (_i x. B)) =/= 0 <-> (A + (_i x. -uB)) =/= 0)
16	4, 7, 15	3bitr4i 200	. . . . 5 |- ((A =/= 0 \/ B =/= 0) <-> (A - (_i x. B)) =/= 0)
17	11, 12	subcli 6523	. . . . . 6 |- (A - (_i x. B)) e. CC
18		divid 6942	. . . . . 6 |- (((A - (_i x. B)) e. CC /\ (A - (_i x. B)) =/= 0) -> ((A - (_i x. B)) / (A - (_i x. B))) = 1)
19	17, 18	mpan 759	. . . . 5 |- ((A - (_i x. B)) =/= 0 -> ((A - (_i x. B)) / (A - (_i x. B))) = 1)
20	16, 19	sylbi 216	. . . 4 |- ((A =/= 0 \/ B =/= 0) -> ((A - (_i x. B)) / (A - (_i x. B))) = 1)
21	20	opreq2d 4898	. . 3 |- ((A =/= 0 \/ B =/= 0) -> ((1 / (A + (_i x. B))) x. ((A - (_i x. B)) / (A - (_i x. B)))) = ((1 / (A + (_i x. B))) x. 1))
22	11, 12	addcli 6473	. . . . 5 |- (A + (_i x. B)) e. CC
23		ax1cn 6422	. . . . . . 7 |- 1 e. CC
24		divmuldiv 6956	. . . . . . 7 |- (((1 e. CC /\ (A - (_i x. B)) e. CC) /\ (((A + (_i x. B)) e. CC /\ (A + (_i x. B)) =/= 0) /\ ((A - (_i x. B)) e. CC /\ (A - (_i x. B)) =/= 0))) -> ((1 / (A + (_i x. B))) x. ((A - (_i x. B)) / (A - (_i x. B)))) = ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))))
25	23, 17, 24	mpanl12 773	. . . . . 6 |- ((((A + (_i x. B)) e. CC /\ (A + (_i x. B)) =/= 0) /\ ((A - (_i x. B)) e. CC /\ (A - (_i x. B)) =/= 0)) -> ((1 / (A + (_i x. B))) x. ((A - (_i x. B)) / (A - (_i x. B)))) = ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))))
26	17, 25	mpanr1 774	. . . . 5 |- ((((A + (_i x. B)) e. CC /\ (A + (_i x. B)) =/= 0) /\ (A - (_i x. B)) =/= 0) -> ((1 / (A + (_i x. B))) x. ((A - (_i x. B)) / (A - (_i x. B)))) = ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))))
27	22, 26	mpanl1 770	. . . 4 |- (((A + (_i x. B)) =/= 0 /\ (A - (_i x. B)) =/= 0) -> ((1 / (A + (_i x. B))) x. ((A - (_i x. B)) / (A - (_i x. B)))) = ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))))
28	1, 2	crne0i 7989	. . . 4 |- ((A =/= 0 \/ B =/= 0) <-> (A + (_i x. B)) =/= 0)
29	27, 28, 16	sylancb 529	. . 3 |- ((A =/= 0 \/ B =/= 0) -> ((1 / (A + (_i x. B))) x. ((A - (_i x. B)) / (A - (_i x. B)))) = ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))))
30	22	recclzi 6903	. . . . 5 |- ((A + (_i x. B)) =/= 0 -> (1 / (A + (_i x. B))) e. CC)
31	28, 30	sylbi 216	. . . 4 |- ((A =/= 0 \/ B =/= 0) -> (1 / (A + (_i x. B))) e. CC)
32		ax1id 6435	. . . 4 |- ((1 / (A + (_i x. B))) e. CC -> ((1 / (A + (_i x. B))) x. 1) = (1 / (A + (_i x. B))))
33	31, 32	syl 12	. . 3 |- ((A =/= 0 \/ B =/= 0) -> ((1 / (A + (_i x. B))) x. 1) = (1 / (A + (_i x. B))))
34	21, 29, 33	3eqtr3rd 1936	. 2 |- ((A =/= 0 \/ B =/= 0) -> (1 / (A + (_i x. B))) = ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))))
35	17	mulid2i 6486	. . 3 |- (1 x. (A - (_i x. B))) = (A - (_i x. B))
36	11, 12	binom2aiOLD 7891	. . . 4 |- ((A + (_i x. B)) x. (A - (_i x. B))) = ((A^2) - ((_i x. B)^2))
37	11	sqcli 7860	. . . . 5 |- (A^2) e. CC
38	12	sqcli 7860	. . . . 5 |- ((_i x. B)^2) e. CC
39	37, 38	negsubi 6538	. . . 4 |- ((A^2) + -u((_i x. B)^2)) = ((A^2) - ((_i x. B)^2))
40	8, 5	sqmuli 7862	. . . . . . . 8 |- ((_i x. B)^2) = ((_i^2) x. (B^2))
41		i2 7982	. . . . . . . . 9 |- (_i^2) = -u1
42	41	opreq1i 4892	. . . . . . . 8 |- ((_i^2) x. (B^2)) = (-u1 x. (B^2))
43	5	sqcli 7860	. . . . . . . . 9 |- (B^2) e. CC
44	23, 43	mulneg1i 6608	. . . . . . . 8 |- (-u1 x. (B^2)) = -u(1 x. (B^2))
45	40, 42, 44	3eqtri 1912	. . . . . . 7 |- ((_i x. B)^2) = -u(1 x. (B^2))
46	45	negeqi 6515	. . . . . 6 |- -u((_i x. B)^2) = -u-u(1 x. (B^2))
47	23, 43	mulcli 6474	. . . . . . 7 |- (1 x. (B^2)) e. CC
48	47	negnegi 6549	. . . . . 6 |- -u-u(1 x. (B^2)) = (1 x. (B^2))
49	43	mulid2i 6486	. . . . . 6 |- (1 x. (B^2)) = (B^2)
50	46, 48, 49	3eqtri 1912	. . . . 5 |- -u((_i x. B)^2) = (B^2)
51	50	opreq2i 4893	. . . 4 |- ((A^2) + -u((_i x. B)^2)) = ((A^2) + (B^2))
52	36, 39, 51	3eqtr2i 1915	. . 3 |- ((A + (_i x. B)) x. (A - (_i x. B))) = ((A^2) + (B^2))
53	35, 52	opreq12i 4894	. 2 |- ((1 x. (A - (_i x. B))) / ((A + (_i x. B)) x. (A - (_i x. B)))) = ((A - (_i x. B)) / ((A^2) + (B^2)))
54	34, 53	syl6eq 1944	1 |- ((A =/= 0 \/ B =/= 0) -> (1 / (A + (_i x. B))) = ((A - (_i x. B)) / ((A^2) + (B^2))))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  2c2 7145  ^cexp 7811

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

Copyright terms: Public domain