Theorem geolim2 12279

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 2238	. . 3 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		geolim2.3	. . . 4 |- ( ph -> M e. NN0 )
3	2	nn0zd 9766	. . 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 9999	. . . . 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 11111	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( A ^ k ) e. CC )
10		eluzelz 9931	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
11	10	adantl 277	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ZZ )
12		0red 8327	. . . . . . . . 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 9768	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. RR )
15	11	zred 9768	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. RR )
16	2	nn0ge0d 9623	. . . . . . . . . 10 |- ( ph -> 0 <_ M )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ M )
18		eluzle 9934	. . . . . . . . . 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 8450	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ x )
21		elnn0z 9657	. . . . . . . 8 |- ( x e. NN0 <-> ( x e. ZZ /\ 0 <_ x ) )
22	11, 20, 21	sylanbrc 421	. . . . . . 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 11111	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( A ^ x ) e. CC )
25		oveq2 6093	. . . . . . . 8 |- ( n = x -> ( A ^ n ) = ( A ^ x ) )
26		eqid 2238	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
27	25, 26	fvmptg 5781	. . . . . . 7 |- ( ( x e. NN0 /\ ( A ^ x ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) = ( A ^ x ) )
28	22, 24, 27	syl2anc 415	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) = ( A ^ x ) )
29	28, 24	eqeltrd 2315	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) e. CC )
30		oveq2 6093	. . . . . . . 8 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31	30, 26	fvmptg 5781	. . . . . . 7 |- ( ( k e. NN0 /\ ( A ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
32	8, 9, 31	syl2anc 415	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
33	32, 4	eqtr4d 2274	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( F ` k ) )
34		addcl 8304	. . . . . 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 10917	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) = seq M ( + , F ) )
37		seqex 10886	. . . . . 6 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
38		ax-1cn 8272	. . . . . . . 8 |- 1 e. CC
39		subcl 8525	. . . . . . . 8 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
40	38, 5, 39	sylancr 418	. . . . . . 7 |- ( ph -> ( 1 - A ) e. CC )
41		1cnd 8342	. . . . . . . 8 |- ( ph -> 1 e. CC )
42		1red 8341	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
43		geolim.2	. . . . . . . . . 10 |- ( ph -> ( abs ` A ) < 1 )
44	5, 42, 43	absltap 12276	. . . . . . . . 9 |- ( ph -> A # 1 )
45		apsym 8934	. . . . . . . . . 10 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
46	5, 41, 45	syl2anc 415	. . . . . . . . 9 |- ( ph -> ( A # 1 <-> 1 # A ) )
47	44, 46	mpbid 147	. . . . . . . 8 |- ( ph -> 1 # A )
48	41, 5, 47	subap0d 8972	. . . . . . 7 |- ( ph -> ( 1 - A ) # 0 )
49	40, 48	recclapd 9111	. . . . . 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 11111	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
53		oveq2 6093	. . . . . . . . 9 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
54	53, 26	fvmptg 5781	. . . . . . . 8 |- ( ( j e. NN0 /\ ( A ^ j ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
55	50, 52, 54	syl2anc 415	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
56	5, 43, 55	geolim 12278	. . . . . 6 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
57		breldmg 4987	. . . . . 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 1383	. . . . 5 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
59		nn0uz 9957	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
60		expcl 10994	. . . . . . . 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 2315	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
63	59, 2, 62	iserex 12105	. . . . 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 2316	. . 3 |- ( ph -> seq M ( + , F ) e. dom ~~> )
66	1, 3, 4, 9, 65	isumclim2 12189	. 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 11111	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
70	67, 69, 31	syl2anc 415	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
71		expcl 10994	. . . . . . . 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 12258	. . . . . 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 9656	. . . . . . 7 |- ( ph -> 0 e. ZZ )
75	59, 74, 70, 72, 56	isumclim 12188	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
76	73, 75	eqtr3d 2273	. . . . 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 12274	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) )
78	77	oveq1d 6100	. . . . 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 2273	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) = ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
80	79	oveq1d 6100	. . 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 11111	. . . . . 6 |- ( ph -> ( A ^ M ) e. CC )
82		subcl 8525	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( A ^ M ) ) e. CC )
83	38, 81, 82	sylancr 418	. . . . 5 |- ( ph -> ( 1 - ( A ^ M ) ) e. CC )
84	41, 83, 40, 48	divsubdirapd 9160	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
85		nncan 8555	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
86	38, 81, 85	sylancr 418	. . . . 5 |- ( ph -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
87	86	oveq1d 6100	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
88	84, 87	eqtr3d 2273	. . 3 |- ( ph -> ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
89	83, 40, 48	divclapd 9120	. . . 4 |- ( ph -> ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) e. CC )
90	1, 3, 32, 9, 64	isumcl 12192	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) e. CC )
91	89, 90	pncan2d 8639	. . 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 2280	. 2 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) = ( ( A ^ M ) / ( 1 - A ) ) )
93	66, 92	breqtrd 4156	1 |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   |-> cmpt 4192   dom cdm 4774   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   # cap 8909   / cdiv 9002   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884   ^cexp 10975   abscabs 11763   ~~> cli 12044   sum_csu 12119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120

This theorem is used by:  geoisum1  12286  geoisum1c  12287  log2tlbndlog2  16082  trilpolemisumle  17087