Theorem geoisumr 8505

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		lt01 6871	. . . . . . 7 |- 0 < 1
2		abscl 8084	. . . . . . . 8 |- (A e. CC -> (abs` A) e. RR)
3		0re 6603	. . . . . . . . 9 |- 0 e. RR
4		1re 6598	. . . . . . . . 9 |- 1 e. RR
5		axlttrn 6673	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR /\ (abs` A) e. RR) -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
6	3, 4, 5	mp3an12 1181	. . . . . . . 8 |- ((abs` A) e. RR -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
7	2, 6	syl 12	. . . . . . 7 |- (A e. CC -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
8	1, 7	mpani 762	. . . . . 6 |- (A e. CC -> (1 < (abs` A) -> 0 < (abs` A)))
9	8	imp 377	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> 0 < (abs` A))
10		absgt0 8145	. . . . . 6 |- (A e. CC -> (A =/= 0 <-> 0 < (abs` A)))
11	10	biimpar 461	. . . . 5 |- ((A e. CC /\ 0 < (abs` A)) -> A =/= 0)
12	9, 11	syldan 516	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> A =/= 0)
13		reccl 6904	. . . 4 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
14	12, 13	syldan 516	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / A) e. CC)
15		ax1cn 6422	. . . . . . 7 |- 1 e. CC
16		absdiv 8111	. . . . . . 7 |- ((1 e. CC /\ A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
17	15, 16	mp3an1 1178	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
18	12, 17	syldan 516	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
19	3, 4, 1	ltleii 6756	. . . . . . 7 |- 0 <_ 1
20	4	absidi 8112	. . . . . . 7 |- (0 <_ 1 -> (abs` 1) = 1)
21	19, 20	ax-mp 7	. . . . . 6 |- (abs` 1) = 1
22	21	opreq1i 4892	. . . . 5 |- ((abs` 1) / (abs` A)) = (1 / (abs` A))
23	18, 22	syl6eq 1944	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = (1 / (abs` A)))
24		recgt1i 7083	. . . . . 6 |- (((abs` A) e. RR /\ 1 < (abs` A)) -> (0 < (1 / (abs` A)) /\ (1 / (abs` A)) < 1))
25	24, 2	sylan 497	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> (0 < (1 / (abs` A)) /\ (1 / (abs` A)) < 1))
26	25	simprd 352	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (abs` A)) < 1)
27	23, 26	eqbrtrd 3357	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) < 1)
28		geoisum 8504	. . 3 |- (((1 / A) e. CC /\ (abs` (1 / A)) < 1) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
29	14, 27, 28	syl11anc 524	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
30		divsubdir 6951	. . . . . . 7 |- ((A e. CC /\ 1 e. CC /\ (A e. CC /\ A =/= 0)) -> ((A - 1) / A) = ((A / A) - (1 / A)))
31	15, 30	mp3an2 1179	. . . . . 6 |- ((A e. CC /\ (A e. CC /\ A =/= 0)) -> ((A - 1) / A) = ((A / A) - (1 / A)))
32	31	anabss5 560	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
33		divid 6942	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
34	33	opreq1d 4897	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A / A) - (1 / A)) = (1 - (1 / A)))
35	32, 34	eqtr2d 1926	. . . 4 |- ((A e. CC /\ A =/= 0) -> (1 - (1 / A)) = ((A - 1) / A))
36	12, 35	syldan 516	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (1 - (1 / A)) = ((A - 1) / A))
37	36	opreq2d 4898	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (1 - (1 / A))) = (1 / ((A - 1) / A)))
38		subcl 6524	. . . . 5 |- ((A e. CC /\ 1 e. CC) -> (A - 1) e. CC)
39	15, 38	mpan2 760	. . . 4 |- (A e. CC -> (A - 1) e. CC)
40	39	adantr 425	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (A - 1) e. CC)
41		ltne 6686	. . . . . 6 |- ((1 e. RR /\ (abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
42	4, 41	mp3an1 1178	. . . . 5 |- (((abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
43	42, 2	sylan 497	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` A) =/= 1)
44		subeq0 6563	. . . . . . . 8 |- ((A e. CC /\ 1 e. CC) -> ((A - 1) = 0 <-> A = 1))
45	15, 44	mpan2 760	. . . . . . 7 |- (A e. CC -> ((A - 1) = 0 <-> A = 1))
46		fveq2 4681	. . . . . . . 8 |- (A = 1 -> (abs` A) = (abs` 1))
47	46, 21	syl6eq 1944	. . . . . . 7 |- (A = 1 -> (abs` A) = 1)
48	45, 47	syl6bi 231	. . . . . 6 |- (A e. CC -> ((A - 1) = 0 -> (abs` A) = 1))
49	48	necon3d 2041	. . . . 5 |- (A e. CC -> ((abs` A) =/= 1 -> (A - 1) =/= 0))
50	49	adantr 425	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> ((abs` A) =/= 1 -> (A - 1) =/= 0))
51	43, 50	mpd 29	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (A - 1) =/= 0)
52		simpl 346	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> A e. CC)
53		recdiv 6967	. . 3 |- ((((A - 1) e. CC /\ (A - 1) =/= 0) /\ (A e. CC /\ A =/= 0)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
54	40, 51, 52, 12, 53	syl22anc 1101	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
55	29, 37, 54	3eqtrd 1929	1 |- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (A / (A - 1)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   - cmin 6445   / cdiv 6447   <_ cle 6448  NN0cn0 6450   < clt 6653  ^cexp 7811  abscabs 8000  sum_csu 8239

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-sup 5664  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-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240

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