Theorem geoisumr 7178

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 7177	. . 3 |- (((1 / A) e. CC /\ (abs` (1 / A)) < 1) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
2		lt01 5653	. . . . . . 7 |- 0 < 1
3		absclt 6768	. . . . . . . 8 |- (A e. CC -> (abs` A) e. RR)
4		0re 5412	. . . . . . . . 9 |- 0 e. RR
5		1re 5407	. . . . . . . . 9 |- 1 e. RR
6		axlttrn 5476	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR /\ (abs` A) e. RR) -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
7	4, 5, 6	mp3an12 903	. . . . . . . 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 696	. . . . . 6 |- (A e. CC -> (1 < (abs` A) -> 0 < (abs` A)))
10	9	imp 350	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> 0 < (abs` A))
11		absgt0t 6831	. . . . . 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 467	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> A =/= 0)
14		recclt 5684	. . . 4 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
15	13, 14	syldan 467	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / A) e. CC)
16		ax1cn 5241	. . . . . . 7 |- 1 e. CC
17		absdivt 6795	. . . . . . 7 |- ((1 e. CC /\ A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
18	16, 17	mp3an1 900	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
19	13, 18	syldan 467	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
20	4, 5, 2	ltlei 5554	. . . . . . 7 |- 0 <_ 1
21	5	absid 6796	. . . . . . 7 |- (0 <_ 1 -> (abs` 1) = 1)
22	20, 21	ax-mp 7	. . . . . 6 |- (abs` 1) = 1
23	22	opreq1i 3956	. . . . 5 |- ((abs` 1) / (abs` A)) = (1 / (abs` A))
24	19, 23	syl6eq 1515	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = (1 / (abs` A)))
25		recgt1it 5848	. . . . . 6 |- (((abs` A) e. RR /\ 1 < (abs` A)) -> (0 < (1 / (abs` A)) /\ (1 / (abs` A)) < 1))
26	25, 3	sylan 448	. . . . 5 |- ((A e. CC /\ 1 < (abs` A)) -> (0 < (1 / (abs` A)) /\ (1 / (abs` A)) < 1))
27	26	pm3.27d 325	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (abs` A)) < 1)
28	24, 27	eqbrtrd 2625	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) < 1)
29	1, 15, 28	sylanc 471	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
30		divsubdirt 5731	. . . . . . 7 |- (((A e. CC /\ 1 e. CC /\ A e. CC) /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
31	16, 30	mp3anl2 908	. . . . . 6 |- (((A e. CC /\ A e. CC) /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
32	31	anabsan 503	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
33		dividt 5722	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
34	33	opreq1d 3960	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A / A) - (1 / A)) = (1 - (1 / A)))
35	32, 34	eqtr2d 1500	. . . 4 |- ((A e. CC /\ A =/= 0) -> (1 - (1 / A)) = ((A - 1) / A))
36	13, 35	syldan 467	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (1 - (1 / A)) = ((A - 1) / A))
37	36	opreq2d 3961	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (1 - (1 / A))) = (1 / ((A - 1) / A)))
38		recdivt 5746	. . 3 |- ((((A - 1) e. CC /\ (A - 1) =/= 0) /\ (A e. CC /\ A =/= 0)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
39		subclt 5339	. . . . 5 |- ((A e. CC /\ 1 e. CC) -> (A - 1) e. CC)
40	16, 39	mpan2 694	. . . 4 |- (A e. CC -> (A - 1) e. CC)
41	40	adantr 389	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> (A - 1) e. CC)
42		ltnet 5488	. . . . . 6 |- ((1 e. RR /\ (abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
43	5, 42	mp3an1 900	. . . . 5 |- (((abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
44	43, 3	sylan 448	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` A) =/= 1)
45		subeq0t 5375	. . . . . . . 8 |- ((A e. CC /\ 1 e. CC) -> ((A - 1) = 0 <-> A = 1))
46	16, 45	mpan2 694	. . . . . . 7 |- (A e. CC -> ((A - 1) = 0 <-> A = 1))
47		fveq2 3709	. . . . . . . 8 |- (A = 1 -> (abs` A) = (abs` 1))
48	47, 22	syl6eq 1515	. . . . . . 7 |- (A = 1 -> (abs` A) = 1)
49	46, 48	syl6bi 214	. . . . . 6 |- (A e. CC -> ((A - 1) = 0 -> (abs` A) = 1))
50	49	necon3d 1596	. . . . 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		pm3.26 319	. . 3 |- ((A e. CC /\ 1 < (abs` A)) -> A e. CC)
54	38, 41, 52, 53, 13	syl2anc 472	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
55	29, 37, 54	3eqtrd 1503	1 |- ((A e. CC /\ 1 < (abs` A)) -> sum_k e. NN0 ((1 / A)^k) = (A / (A - 1)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206  1c1 5207   - cmin 5264   / cdiv 5266   <_ cle 5267  NN0cn0 5269   < clt 5458  ^cexp 6500  abscabs 6681  sum_csu 6917

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-fl 6172  df-seq1 6245  df-shft 6278  df-uz 6350  df-fz 6400  df-seqz 6465  df-seq0 6466  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-clim 6913  df-sum 6918

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