Theorem geolim2 14602

Description: The partial sums in the geometric series A ^ M + A ^ ( M + 1 )... converge to ( ( A ^ M ) / ( 1 - A ) ). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)

Hypotheses

Ref	Expression
geolim.1	|- ( ph -> A e. CC )
geolim.2	|- ( ph -> ( abs ` A ) < 1 )
geolim2.3	|- ( ph -> M e. NN0 )
geolim2.4	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( A ^ k ) )

Assertion

Ref	Expression
geolim2	|- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

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

Proof of Theorem geolim2

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		geolim2.3	. . . 4 |- ( ph -> M e. NN0 )
3	2	nn0zd 11480	. . 3 |- ( ph -> M e. ZZ )
4		geolim2.4	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( A ^ k ) )
5		geolim.1	. . . . 5 |- ( ph -> A e. CC )
6	5	adantr 481	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
7		eluznn0 11757	. . . . 5 |- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
8	2, 7	sylan 488	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
9	6, 8	expcld 13008	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( A ^ k ) e. CC )
10		oveq2 6658	. . . . . . . 8 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
11		eqid 2622	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
12		ovex 6678	. . . . . . . 8 |- ( A ^ k ) e. _V
13	10, 11, 12	fvmpt 6282	. . . . . . 7 |- ( k e. NN0 -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
14	8, 13	syl 17	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
15	14, 4	eqtr4d 2659	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( F ` k ) )
16	3, 15	seqfeq 12826	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) = seq M ( + , F ) )
17		geolim.2	. . . . . . 7 |- ( ph -> ( abs ` A ) < 1 )
18		oveq2 6658	. . . . . . . . 9 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
19		ovex 6678	. . . . . . . . 9 |- ( A ^ j ) e. _V
20	18, 11, 19	fvmpt 6282	. . . . . . . 8 |- ( j e. NN0 -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
21	20	adantl 482	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
22	5, 17, 21	geolim 14601	. . . . . 6 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
23		seqex 12803	. . . . . . 7 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
24		ovex 6678	. . . . . . 7 |- ( 1 / ( 1 - A ) ) e. _V
25	23, 24	breldm 5329	. . . . . 6 |- ( seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
26	22, 25	syl 17	. . . . 5 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
27		nn0uz 11722	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
28		expcl 12878	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ j ) e. CC )
29	5, 28	sylan 488	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
30	21, 29	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
31	27, 2, 30	iserex 14387	. . . . 5 |- ( ph -> ( seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> <-> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> ) )
32	26, 31	mpbid 222	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
33	16, 32	eqeltrrd 2702	. . 3 |- ( ph -> seq M ( + , F ) e. dom ~~> )
34	1, 3, 4, 9, 33	isumclim2 14489	. 2 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) )
35	13	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
36		expcl 12878	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
37	5, 36	sylan 488	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
38	27, 1, 2, 35, 37, 26	isumsplit 14572	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
39		0zd 11389	. . . . . . 7 |- ( ph -> 0 e. ZZ )
40	27, 39, 35, 37, 22	isumclim 14488	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
41	38, 40	eqtr3d 2658	. . . . 5 |- ( ph -> ( sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) = ( 1 / ( 1 - A ) ) )
42		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
43	42	ltnri 10146	. . . . . . . . . 10 |- -. 1 < 1
44		fveq2 6191	. . . . . . . . . . . 12 |- ( A = 1 -> ( abs ` A ) = ( abs ` 1 ) )
45		abs1 14037	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
46	44, 45	syl6eq 2672	. . . . . . . . . . 11 |- ( A = 1 -> ( abs ` A ) = 1 )
47	46	breq1d 4663	. . . . . . . . . 10 |- ( A = 1 -> ( ( abs ` A ) < 1 <-> 1 < 1 ) )
48	43, 47	mtbiri 317	. . . . . . . . 9 |- ( A = 1 -> -. ( abs ` A ) < 1 )
49	48	necon2ai 2823	. . . . . . . 8 |- ( ( abs ` A ) < 1 -> A =/= 1 )
50	17, 49	syl 17	. . . . . . 7 |- ( ph -> A =/= 1 )
51	5, 50, 2	geoser 14599	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) )
52	51	oveq1d 6665	. . . . 5 |- ( ph -> ( sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) = ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
53	41, 52	eqtr3d 2658	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) = ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
54	53	oveq1d 6665	. . 3 |- ( ph -> ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = ( ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
55		1cnd 10056	. . . . 5 |- ( ph -> 1 e. CC )
56		ax-1cn 9994	. . . . . 6 |- 1 e. CC
57	5, 2	expcld 13008	. . . . . 6 |- ( ph -> ( A ^ M ) e. CC )
58		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( A ^ M ) ) e. CC )
59	56, 57, 58	sylancr 695	. . . . 5 |- ( ph -> ( 1 - ( A ^ M ) ) e. CC )
60		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
61	56, 5, 60	sylancr 695	. . . . 5 |- ( ph -> ( 1 - A ) e. CC )
62	50	necomd 2849	. . . . . 6 |- ( ph -> 1 =/= A )
63		subeq0 10307	. . . . . . . 8 |- ( ( 1 e. CC /\ A e. CC ) -> ( ( 1 - A ) = 0 <-> 1 = A ) )
64	56, 5, 63	sylancr 695	. . . . . . 7 |- ( ph -> ( ( 1 - A ) = 0 <-> 1 = A ) )
65	64	necon3bid 2838	. . . . . 6 |- ( ph -> ( ( 1 - A ) =/= 0 <-> 1 =/= A ) )
66	62, 65	mpbird 247	. . . . 5 |- ( ph -> ( 1 - A ) =/= 0 )
67	55, 59, 61, 66	divsubdird 10840	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
68		nncan 10310	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
69	56, 57, 68	sylancr 695	. . . . 5 |- ( ph -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
70	69	oveq1d 6665	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
71	67, 70	eqtr3d 2658	. . 3 |- ( ph -> ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
72	59, 61, 66	divcld 10801	. . . 4 |- ( ph -> ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) e. CC )
73	1, 3, 14, 9, 32	isumcl 14492	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) e. CC )
74	72, 73	pncan2d 10394	. . 3 |- ( ph -> ( ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = sum_ k e. ( ZZ>= ` M ) ( A ^ k ) )
75	54, 71, 74	3eqtr3rd 2665	. 2 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) = ( ( A ^ M ) / ( 1 - A ) ) )
76	34, 75	breqtrd 4679	1 |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < 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:  geoisum1  14610  geoisum1c  14611  rpnnen2lem3  14945  rpnnen2lem9  14951  abelthlem7  24192  log2tlbnd  24672  geomcau  33555  stirlinglem10  40300