Theorem geolim 14601

Description: The partial sums in the infinite series 1 + A ^ 1 + A ^ 2... converge to ( 1 / ( 1 - A ) ). (Contributed by NM, 15-May-2006.)

Hypotheses

Ref	Expression
geolim.1	|- ( ph -> A e. CC )
geolim.2	|- ( ph -> ( abs ` A ) < 1 )
geolim.3	|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( A ^ k ) )

Assertion

Ref	Expression
geolim	|- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )

Distinct variable groups:   A, k   k, F   ph, k

Proof of Theorem geolim

Dummy variables j n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 11722	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 11389	. . 3 |- ( ph -> 0 e. ZZ )
3		geolim.1	. . . . . 6 |- ( ph -> A e. CC )
4		geolim.2	. . . . . 6 |- ( ph -> ( abs ` A ) < 1 )
5	3, 4	expcnv 14596	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
6		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
7		subcl 10280	. . . . . . 7 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
8	6, 3, 7	sylancr 695	. . . . . 6 |- ( ph -> ( 1 - A ) e. CC )
9		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
10	9	ltnri 10146	. . . . . . . . . . 11 |- -. 1 < 1
11		fveq2 6191	. . . . . . . . . . . . 13 |- ( A = 1 -> ( abs ` A ) = ( abs ` 1 ) )
12		abs1 14037	. . . . . . . . . . . . 13 |- ( abs ` 1 ) = 1
13	11, 12	syl6eq 2672	. . . . . . . . . . . 12 |- ( A = 1 -> ( abs ` A ) = 1 )
14	13	breq1d 4663	. . . . . . . . . . 11 |- ( A = 1 -> ( ( abs ` A ) < 1 <-> 1 < 1 ) )
15	10, 14	mtbiri 317	. . . . . . . . . 10 |- ( A = 1 -> -. ( abs ` A ) < 1 )
16	15	necon2ai 2823	. . . . . . . . 9 |- ( ( abs ` A ) < 1 -> A =/= 1 )
17	4, 16	syl 17	. . . . . . . 8 |- ( ph -> A =/= 1 )
18	17	necomd 2849	. . . . . . 7 |- ( ph -> 1 =/= A )
19		subeq0 10307	. . . . . . . . 9 |- ( ( 1 e. CC /\ A e. CC ) -> ( ( 1 - A ) = 0 <-> 1 = A ) )
20	6, 3, 19	sylancr 695	. . . . . . . 8 |- ( ph -> ( ( 1 - A ) = 0 <-> 1 = A ) )
21	20	necon3bid 2838	. . . . . . 7 |- ( ph -> ( ( 1 - A ) =/= 0 <-> 1 =/= A ) )
22	18, 21	mpbird 247	. . . . . 6 |- ( ph -> ( 1 - A ) =/= 0 )
23	3, 8, 22	divcld 10801	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
24		nn0ex 11298	. . . . . . 7 |- NN0 e. _V
25	24	mptex 6486	. . . . . 6 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) e. _V
26	25	a1i 11	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) e. _V )
27		oveq2 6658	. . . . . . . 8 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
28		eqid 2622	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
29		ovex 6678	. . . . . . . 8 |- ( A ^ j ) e. _V
30	27, 28, 29	fvmpt 6282	. . . . . . 7 |- ( j e. NN0 -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
31	30	adantl 482	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
32		expcl 12878	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ j ) e. CC )
33	3, 32	sylan 488	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
34	31, 33	eqeltrd 2701	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
35		expp1 12867	. . . . . . . . . 10 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( ( A ^ j ) x. A ) )
36	3, 35	sylan 488	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( ( A ^ j ) x. A ) )
37	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ j e. NN0 ) -> A e. CC )
38	33, 37	mulcomd 10061	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ j ) x. A ) = ( A x. ( A ^ j ) ) )
39	36, 38	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( A x. ( A ^ j ) ) )
40	39	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) )
41	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( 1 - A ) e. CC )
42	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( 1 - A ) =/= 0 )
43	37, 33, 41, 42	div23d 10838	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
44	40, 43	eqtrd 2656	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
45		oveq1 6657	. . . . . . . . . 10 |- ( n = j -> ( n + 1 ) = ( j + 1 ) )
46	45	oveq2d 6666	. . . . . . . . 9 |- ( n = j -> ( A ^ ( n + 1 ) ) = ( A ^ ( j + 1 ) ) )
47	46	oveq1d 6665	. . . . . . . 8 |- ( n = j -> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
48		eqid 2622	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) = ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) )
49		ovex 6678	. . . . . . . 8 |- ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. _V
50	47, 48, 49	fvmpt 6282	. . . . . . 7 |- ( j e. NN0 -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
51	50	adantl 482	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
52	31	oveq2d 6666	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A / ( 1 - A ) ) x. ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
53	44, 51, 52	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A / ( 1 - A ) ) x. ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) ) )
54	1, 2, 5, 23, 26, 34, 53	climmulc2 14367	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> ( ( A / ( 1 - A ) ) x. 0 ) )
55	23	mul01d 10235	. . . 4 |- ( ph -> ( ( A / ( 1 - A ) ) x. 0 ) = 0 )
56	54, 55	breqtrd 4679	. . 3 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> 0 )
57	8, 22	reccld 10794	. . 3 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
58		seqex 12803	. . . 4 |- seq 0 ( + , F ) e. _V
59	58	a1i 11	. . 3 |- ( ph -> seq 0 ( + , F ) e. _V )
60		peano2nn0 11333	. . . . . 6 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
61		expcl 12878	. . . . . 6 |- ( ( A e. CC /\ ( j + 1 ) e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
62	3, 60, 61	syl2an 494	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
63	62, 41, 42	divcld 10801	. . . 4 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
64	51, 63	eqeltrd 2701	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) e. CC )
65		nn0cn 11302	. . . . . . . 8 |- ( j e. NN0 -> j e. CC )
66	65	adantl 482	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. CC )
67		pncan 10287	. . . . . . 7 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
68	66, 6, 67	sylancl 694	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( j + 1 ) - 1 ) = j )
69	68	oveq2d 6666	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( 0 ... ( ( j + 1 ) - 1 ) ) = ( 0 ... j ) )
70	69	sumeq1d 14431	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = sum_ k e. ( 0 ... j ) ( A ^ k ) )
71	6	a1i 11	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> 1 e. CC )
72	71, 62, 41, 42	divsubdird 10840	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) ) )
73	17	adantr 481	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> A =/= 1 )
74	60	adantl 482	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( j + 1 ) e. NN0 )
75	37, 73, 74	geoser 14599	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) )
76	51	oveq2d 6666	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( 1 / ( 1 - A ) ) - ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) ) = ( ( 1 / ( 1 - A ) ) - ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) ) )
77	72, 75, 76	3eqtr4d 2666	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 / ( 1 - A ) ) - ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) ) )
78		simpll 790	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> ph )
79		elfznn0 12433	. . . . . . 7 |- ( k e. ( 0 ... j ) -> k e. NN0 )
80	79	adantl 482	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> k e. NN0 )
81		geolim.3	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( A ^ k ) )
82	78, 80, 81	syl2anc 693	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> ( F ` k ) = ( A ^ k ) )
83		simpr 477	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> j e. NN0 )
84	83, 1	syl6eleq 2711	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> j e. ( ZZ>= ` 0 ) )
85	78, 3	syl 17	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> A e. CC )
86	85, 80	expcld 13008	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> ( A ^ k ) e. CC )
87	82, 84, 86	fsumser 14461	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( A ^ k ) = ( seq 0 ( + , F ) ` j ) )
88	70, 77, 87	3eqtr3rd 2665	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( seq 0 ( + , F ) ` j ) = ( ( 1 / ( 1 - A ) ) - ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) ) )
89	1, 2, 56, 57, 59, 64, 88	climsubc2 14369	. 2 |- ( ph -> seq 0 ( + , F ) ~~> ( ( 1 / ( 1 - A ) ) - 0 ) )
90	57	subid1d 10381	. 2 |- ( ph -> ( ( 1 / ( 1 - A ) ) - 0 ) = ( 1 / ( 1 - A ) ) )
91	89, 90	breqtrd 4679	1 |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  geolim2  14602  georeclim  14603  geomulcvg  14607  geoisum  14608  cvgrat  14615  eflegeo  14851  geolim3  24094  abelthlem5  24189  logtayllem  24405  zetacvg  24741  knoppcnlem6  32488