Theorem geoisumr 7366

Description: The infinite sum of reciprocals 1 + (1 / A)^1 + (1 / A)^2 ... is A / (A - 1). (Contributed by rpenner, 3-Nov-2007.)

Assertion

Ref	Expression
geoisumr	|- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (A / (A - 1)))

Distinct variable group:   A,k

Proof of Theorem geoisumr

Step	Hyp	Ref	Expression
1		geoisum 7365	. . 3 |- (((1 / A) e. CC /\ (abs` (1 / A)) < 1) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
2		lt01 5769	. . . . . . 7 |- 0 < 1
3		abscl 6956	. . . . . . . 8 |- (A e. CC -> (abs` A) e. RR)
4		0re 5529	. . . . . . . . 9 |- 0 e. RR
5		1re 5524	. . . . . . . . 9 |- 1 e. RR
6		axlttrn 5593	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR /\ (abs` A) e. RR) -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
7	4, 5, 6	mp3an12 909	. . . . . . . 8 |- ((abs` A) e. RR -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
8	3, 7	syl 10	. . . . . . 7 |- (A e. CC -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
9	2, 8	mpani 701	. . . . . 6 |- (A e. CC -> (1 < (abs` A) -> 0 < (abs` A)))
10	9	imp 348	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> 0 < (abs` A))
11		absgt0 7016	. . . . . 6 |- (A e. CC -> (A =/= 0 <-> 0 < (abs` A)))
12	11	biimpar 417	. . . . 5 |- ((A e. CC /\ 0 < (abs` A)) -> A =/= 0)
13	10, 12	syldan 469	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> A =/= 0)
14		reccl 5804	. . . 4 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
15	13, 14	syldan 469	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / A) e. CC)
16		ax1cn 5358	. . . . . . 7 |- 1 e. CC
17		absdiv 6983	. . . . . . 7 |- ((1 e. CC /\ A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
18	16, 17	mp3an1 906	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
19	13, 18	syldan 469	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
20	4, 5, 2	ltleii 5670	. . . . . . 7 |- 0 <_ 1
21	5	absidi 6984	. . . . . . 7 |- (0 <_ 1 -> (abs` 1) = 1)
22	20, 21	ax-mp 7	. . . . . 6 |- (abs` 1) = 1
23	22	opreq1i 4047	. . . . 5 |- ((abs` 1) / (abs` A)) = (1 / (abs` A))
24	19, 23	syl6eq 1560	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = (1 / (abs` A)))
25		recgt1i 5987	. . . . . 6 |- (((abs` A) e. RR /\ 1 < (abs` A)) -> (0 < (1 / (abs` A)) /\ (1 / (abs` A)) < 1))
26	25, 3	sylan 450	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> (0 < (1 / (abs` A)) /\ (1 / (abs` A)) < 1))
27	26	pm3.27d 323	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (abs` A)) < 1)
28	24, 27	eqbrtrd 2685	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) < 1)
29	1, 15, 28	sylanc 473	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
30		divsubdirOLD 5855	. . . . . . 7 |- (((A e. CC /\ 1 e. CC /\ A e. CC) /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
31	16, 30	mp3anl2 914	. . . . . 6 |- (((A e. CC /\ A e. CC) /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
32	31	anabsan 506	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
33		divid 5845	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
34	33	opreq1d 4051	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A / A) - (1 / A)) = (1 - (1 / A)))
35	32, 34	eqtr2d 1545	. . . 4 |- ((A e. CC /\ A =/= 0) -> (1 - (1 / A)) = ((A - 1) / A))
36	13, 35	syldan 469	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (1 - (1 / A)) = ((A - 1) / A))
37	36	opreq2d 4052	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (1 - (1 / A))) = (1 / ((A - 1) / A)))
38		recdiv 5873	. . 3 |- ((((A - 1) e. CC /\ (A - 1) =/= 0) /\ (A e. CC /\ A =/= 0)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
39		subcl 5456	. . . . 5 |- ((A e. CC /\ 1 e. CC) -> (A - 1) e. CC)
40	16, 39	mpan2 699	. . . 4 |- (A e. CC -> (A - 1) e. CC)
41	40	adantr 389	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (A - 1) e. CC)
42		ltne 5605	. . . . . 6 |- ((1 e. RR /\ (abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
43	5, 42	mp3an1 906	. . . . 5 |- (((abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
44	43, 3	sylan 450	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` A) =/= 1)
45		subeq0 5492	. . . . . . . 8 |- ((A e. CC /\ 1 e. CC) -> ((A - 1) = 0 <-> A = 1))
46	16, 45	mpan2 699	. . . . . . 7 |- (A e. CC -> ((A - 1) = 0 <-> A = 1))
47		fveq2 3800	. . . . . . . 8 |- (A = 1 -> (abs` A) = (abs` 1))
48	47, 22	syl6eq 1560	. . . . . . 7 |- (A = 1 -> (abs` A) = 1)
49	46, 48	syl6bi 212	. . . . . 6 |- (A e. CC -> ((A - 1) = 0 -> (abs` A) = 1))
50	49	necon3d 1641	. . . . 5 |- (A e. CC -> ((abs` A) =/= 1 -> (A - 1) =/= 0))
51	50	adantr 389	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> ((abs` A) =/= 1 -> (A - 1) =/= 0))
52	44, 51	mpd 26	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (A - 1) =/= 0)
53		pm3.26 317	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> A e. CC)
54	38, 41, 52, 53, 13	syl2anc 474	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
55	29, 37, 54	3eqtrd 1548	1 |- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (A / (A - 1)))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990   =/= wne 1622   class class class wbr 2669  ` cfv 3237  (class class class)co 4039  CCcc 5321  RRcr 5322  0cc0 5323  1c1 5324   - cmin 5381   / cdiv 5383   <_ cle 5384  NN0cn0 5386   < clt 5575  ^cexp 6691  abscabs 6873  sum_csu 7102

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920  ax-inf2 4711

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 779  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-nel 1625  df-ral 1687  df-rex 1688  df-reu 1689  df-rab 1690  df-v 1850  df-sbc 1979  df-csb 2044  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-pss 2099  df-nul 2325  df-if 2407  df-pw 2447  df-sn 2457  df-pr 2458  df-tp 2460  df-op 2461  df-uni 2552  df-int 2582  df-iun 2616  df-br 2670  df-opab 2718  df-tr 2732  df-eprel 2886  df-id 2889  df-po 2894  df-so 2904  df-fr 2972  df-we 2989  df-ord 3006  df-on 3007  df-lim 3008  df-suc 3009  df-om 3193  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-rdg 4008  df-opr 4041  df-oprab 4042  df-1st 4157  df-2nd 4158  df-1o 4217  df-oadd 4219  df-omul 4220  df-er 4345  df-ec 4347  df-qs 4350  df-en 4455  df-dom 4456  df-sdom 4457  df-sup 4658  df-ni 5089  df-pli 5090  df-mi 5091  df-lti 5092  df-plpq 5124  df-mpq 5125  df-enq 5126  df-nq 5127  df-plq 5128  df-mq 5129  df-rq 5130  df-ltq 5131  df-1q 5132  df-np 5175  df-1p 5176  df-plp 5177  df-mp 5178  df-ltp 5179  df-plpr 5253  df-mpr 5254  df-enr 5255  df-nr 5256  df-plr 5257  df-mr 5258  df-ltr 5259  df-0r 5260  df-1r 5261  df-m1r 5262  df-c 5329  df-0 5330  df-1 5331  df-i 5332  df-r 5333  df-plus 5334  df-mul 5335  df-lt 5336  df-sub 5445  df-neg 5447  df-pnf 5576  df-mnf 5577  df-xr 5578  df-ltxr 5579  df-le 5580  df-div 5789  df-n 6012  df-2 6058  df-n0 6210  df-z 6246  df-fl 6363  df-uz 6478  df-fz 6528  df-seq1 6601  df-shft 6634  df-seqz 6656  df-seq0 6657  df-exp 6692  df-sqr 6793  df-re 6874  df-im 6875  df-cj 6876  df-abs 6877  df-clim 7098  df-sum 7103

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