Theorem geoisumr 7243

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 7242	. . 3 |- (((1 / A) e. CC /\ (abs` (1 / A)) < 1) -> sum_k e. NN0 ((1 / A)^k) = (1 / (1 - (1 / A))))
2		lt01 5680	. . . . . . 7 |- 0 < 1
3		absclt 6833	. . . . . . . 8 |- (A e. CC -> (abs` A) e. RR)
4		0re 5440	. . . . . . . . 9 |- 0 e. RR
5		1re 5435	. . . . . . . . 9 |- 1 e. RR
6		axlttrn 5504	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR /\ (abs` A) e. RR) -> ((0 < 1 /\ 1 < (abs` A)) -> 0 < (abs` A)))
7	4, 5, 6	mp3an12 906	. . . . . . . 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 698	. . . . . 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 6893	. . . . . 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 5715	. . . 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 5269	. . . . . . 7 |- 1 e. CC
17		absdivt 6860	. . . . . . 7 |- ((1 e. CC /\ A e. CC /\ A =/= 0) -> (abs` (1 / A)) = ((abs` 1) / (abs` A)))
18	16, 17	mp3an1 903	. . . . . 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 5581	. . . . . . 7 |- 0 <_ 1
21	5	absid 6861	. . . . . . 7 |- (0 <_ 1 -> (abs` 1) = 1)
22	20, 21	ax-mp 7	. . . . . 6 |- (abs` 1) = 1
23	22	opreq1i 3971	. . . . 5 |- ((abs` 1) / (abs` A)) = (1 / (abs` A))
24	19, 23	syl6eq 1523	. . . 4 |- ((A e. CC /\ 1 < (abs` A)) -> (abs` (1 / A)) = (1 / (abs` A)))
25		recgt1it 5900	. . . . . 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 2635	. . 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		divsubdirtOLD 5775	. . . . . . 7 |- (((A e. CC /\ 1 e. CC /\ A e. CC) /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
31	16, 30	mp3anl2 911	. . . . . 6 |- (((A e. CC /\ A e. CC) /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
32	31	anabsan 504	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A - 1) / A) = ((A / A) - (1 / A)))
33		dividt 5766	. . . . . 6 |- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
34	33	opreq1d 3975	. . . . 5 |- ((A e. CC /\ A =/= 0) -> ((A / A) - (1 / A)) = (1 - (1 / A)))
35	32, 34	eqtr2d 1508	. . . 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 3976	. 2 |- ((A e. CC /\ 1 < (abs` A)) -> (1 / (1 - (1 / A))) = (1 / ((A - 1) / A)))
38		recdivt 5790	. . 3 |- ((((A - 1) e. CC /\ (A - 1) =/= 0) /\ (A e. CC /\ A =/= 0)) -> (1 / ((A - 1) / A)) = (A / (A - 1)))
39		subclt 5367	. . . . 5 |- ((A e. CC /\ 1 e. CC) -> (A - 1) e. CC)
40	16, 39	mpan2 696	. . . 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 5516	. . . . . 6 |- ((1 e. RR /\ (abs` A) e. RR /\ 1 < (abs` A)) -> (abs` A) =/= 1)
43	5, 42	mp3an1 903	. . . . 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 5403	. . . . . . . 8 |- ((A e. CC /\ 1 e. CC) -> ((A - 1) = 0 <-> A = 1))
46	16, 45	mpan2 696	. . . . . . 7 |- (A e. CC -> ((A - 1) = 0 <-> A = 1))
47		fveq2 3724	. . . . . . . 8 |- (A = 1 -> (abs` A) = (abs` 1))
48	47, 22	syl6eq 1523	. . . . . . 7 |- (A = 1 -> (abs` A) = 1)
49	46, 48	syl6bi 214	. . . . . 6 |- (A e. CC -> ((A - 1) = 0 -> (abs` A) = 1))
50	49	necon3d 1604	. . . . 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 1511	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 956   e. wcel 958   =/= wne 1585   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234  1c1 5235   - cmin 5292   / cdiv 5294   <_ cle 5295  NN0cn0 5297   < clt 5486  ^cexp 6568  abscabs 6750  sum_csu 6979

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-n0 6100  df-z 6136  df-fl 6224  df-seq1 6308  df-shft 6341  df-uz 6418  df-fz 6468  df-seqz 6533  df-seq0 6534  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-clim 6975  df-sum 6980

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