Theorem geolim2 11658

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 x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . 3 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		geolim2.3	. . . 4 |- ( ph -> M e. NN0 )
3	2	nn0zd 9440	. . 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 276	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
7		eluznn0 9667	. . . . 5 |- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
8	2, 7	sylan 283	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
9	6, 8	expcld 10747	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( A ^ k ) e. CC )
10		eluzelz 9604	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
11	10	adantl 277	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ZZ )
12		0red 8022	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 e. RR )
13	3	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. ZZ )
14	13	zred 9442	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. RR )
15	11	zred 9442	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. RR )
16	2	nn0ge0d 9299	. . . . . . . . . 10 |- ( ph -> 0 <_ M )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ M )
18		eluzle 9607	. . . . . . . . . 10 |- ( x e. ( ZZ>= ` M ) -> M <_ x )
19	18	adantl 277	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M <_ x )
20	12, 14, 15, 17, 19	letrd 8145	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ x )
21		elnn0z 9333	. . . . . . . 8 |- ( x e. NN0 <-> ( x e. ZZ /\ 0 <_ x ) )
22	11, 20, 21	sylanbrc 417	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. NN0 )
23	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A e. CC )
24	23, 22	expcld 10747	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( A ^ x ) e. CC )
25		oveq2 5927	. . . . . . . 8 |- ( n = x -> ( A ^ n ) = ( A ^ x ) )
26		eqid 2193	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
27	25, 26	fvmptg 5634	. . . . . . 7 |- ( ( x e. NN0 /\ ( A ^ x ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) = ( A ^ x ) )
28	22, 24, 27	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) = ( A ^ x ) )
29	28, 24	eqeltrd 2270	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) e. CC )
30		oveq2 5927	. . . . . . . 8 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31	30, 26	fvmptg 5634	. . . . . . 7 |- ( ( k e. NN0 /\ ( A ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
32	8, 9, 31	syl2anc 411	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
33	32, 4	eqtr4d 2229	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( F ` k ) )
34		addcl 7999	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
35	34	adantl 277	. . . . 5 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
36	3, 29, 33, 35	seq3feq 10554	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) = seq M ( + , F ) )
37		seqex 10523	. . . . . 6 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
38		ax-1cn 7967	. . . . . . . 8 |- 1 e. CC
39		subcl 8220	. . . . . . . 8 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
40	38, 5, 39	sylancr 414	. . . . . . 7 |- ( ph -> ( 1 - A ) e. CC )
41		1cnd 8037	. . . . . . . 8 |- ( ph -> 1 e. CC )
42		1red 8036	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
43		geolim.2	. . . . . . . . . 10 |- ( ph -> ( abs ` A ) < 1 )
44	5, 42, 43	absltap 11655	. . . . . . . . 9 |- ( ph -> A # 1 )
45		apsym 8627	. . . . . . . . . 10 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
46	5, 41, 45	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( A # 1 <-> 1 # A ) )
47	44, 46	mpbid 147	. . . . . . . 8 |- ( ph -> 1 # A )
48	41, 5, 47	subap0d 8665	. . . . . . 7 |- ( ph -> ( 1 - A ) # 0 )
49	40, 48	recclapd 8802	. . . . . 6 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
50		simpr 110	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> j e. NN0 )
51	5	adantr 276	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> A e. CC )
52	51, 50	expcld 10747	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
53		oveq2 5927	. . . . . . . . 9 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
54	53, 26	fvmptg 5634	. . . . . . . 8 |- ( ( j e. NN0 /\ ( A ^ j ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
55	50, 52, 54	syl2anc 411	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
56	5, 43, 55	geolim 11657	. . . . . 6 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
57		breldmg 4869	. . . . . 6 |- ( ( seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V /\ ( 1 / ( 1 - A ) ) e. CC /\ seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) ) -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
58	37, 49, 56, 57	mp3an2i 1353	. . . . 5 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
59		nn0uz 9630	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
60		expcl 10631	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ j ) e. CC )
61	5, 60	sylan 283	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
62	55, 61	eqeltrd 2270	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
63	59, 2, 62	iserex 11485	. . . . 5 |- ( ph -> ( seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> <-> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> ) )
64	58, 63	mpbid 147	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
65	36, 64	eqeltrrd 2271	. . 3 |- ( ph -> seq M ( + , F ) e. dom ~~> )
66	1, 3, 4, 9, 65	isumclim2 11568	. 2 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) )
67		simpr 110	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
68	5	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> A e. CC )
69	68, 67	expcld 10747	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
70	67, 69, 31	syl2anc 411	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
71		expcl 10631	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
72	5, 71	sylan 283	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
73	59, 1, 2, 70, 72, 58	isumsplit 11637	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
74		0zd 9332	. . . . . . 7 |- ( ph -> 0 e. ZZ )
75	59, 74, 70, 72, 56	isumclim 11567	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
76	73, 75	eqtr3d 2228	. . . . 5 |- ( ph -> ( sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) = ( 1 / ( 1 - A ) ) )
77	5, 44, 2	geoserap 11653	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) )
78	77	oveq1d 5934	. . . . 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 ) ) )
79	76, 78	eqtr3d 2228	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) = ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
80	79	oveq1d 5934	. . 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 ) ) ) )
81	5, 2	expcld 10747	. . . . . 6 |- ( ph -> ( A ^ M ) e. CC )
82		subcl 8220	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( A ^ M ) ) e. CC )
83	38, 81, 82	sylancr 414	. . . . 5 |- ( ph -> ( 1 - ( A ^ M ) ) e. CC )
84	41, 83, 40, 48	divsubdirapd 8851	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
85		nncan 8250	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
86	38, 81, 85	sylancr 414	. . . . 5 |- ( ph -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
87	86	oveq1d 5934	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
88	84, 87	eqtr3d 2228	. . 3 |- ( ph -> ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
89	83, 40, 48	divclapd 8811	. . . 4 |- ( ph -> ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) e. CC )
90	1, 3, 32, 9, 64	isumcl 11571	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) e. CC )
91	89, 90	pncan2d 8334	. . 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 ) )
92	80, 88, 91	3eqtr3rd 2235	. 2 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) = ( ( A ^ M ) / ( 1 - A ) ) )
93	66, 92	breqtrd 4056	1 |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   |-> cmpt 4091   dom cdm 4660   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   < clt 8056   <_ cle 8057   - cmin 8192   # cap 8602   / cdiv 8693   NN0cn0 9243   ZZcz 9320   ZZ>=cuz 9595   ...cfz 10077   seqcseq 10521   ^cexp 10612   abscabs 11144   ~~> cli 11424   sum_csu 11499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500

This theorem is referenced by:  geoisum1  11665  geoisum1c  11666  trilpolemisumle  15598