Theorem geolim2 12257

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 9745	. . 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 9978	. . . . 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 11089	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( A ^ k ) e. CC )
10		eluzelz 9910	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
11	10	adantl 277	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ZZ )
12		0red 8317	. . . . . . . . 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 9747	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. RR )
15	11	zred 9747	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. RR )
16	2	nn0ge0d 9602	. . . . . . . . . 10 |- ( ph -> 0 <_ M )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ M )
18		eluzle 9913	. . . . . . . . . 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 8440	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ x )
21		elnn0z 9636	. . . . . . . 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 11089	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( A ^ x ) e. CC )
25		oveq2 6083	. . . . . . . 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 5775	. . . . . . 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 6083	. . . . . . . 8 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31	30, 26	fvmptg 5775	. . . . . . 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 8294	. . . . . 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 10895	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) = seq M ( + , F ) )
37		seqex 10864	. . . . . 6 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
38		ax-1cn 8262	. . . . . . . 8 |- 1 e. CC
39		subcl 8515	. . . . . . . 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 8332	. . . . . . . 8 |- ( ph -> 1 e. CC )
42		1red 8331	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
43		geolim.2	. . . . . . . . . 10 |- ( ph -> ( abs ` A ) < 1 )
44	5, 42, 43	absltap 12254	. . . . . . . . 9 |- ( ph -> A # 1 )
45		apsym 8924	. . . . . . . . . 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 8962	. . . . . . 7 |- ( ph -> ( 1 - A ) # 0 )
49	40, 48	recclapd 9101	. . . . . 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 11089	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
53		oveq2 6083	. . . . . . . . 9 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
54	53, 26	fvmptg 5775	. . . . . . . 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 12256	. . . . . 6 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
57		breldmg 4982	. . . . . 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 9936	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
60		expcl 10972	. . . . . . . 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 12083	. . . . 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 12167	. 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 11089	. . . . . . . 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 10972	. . . . . . . 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 12236	. . . . . 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 9635	. . . . . . 7 |- ( ph -> 0 e. ZZ )
75	59, 74, 70, 72, 56	isumclim 12166	. . . . . 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 12252	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) )
78	77	oveq1d 6090	. . . . 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 6090	. . 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 11089	. . . . . 6 |- ( ph -> ( A ^ M ) e. CC )
82		subcl 8515	. . . . . 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 9150	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
85		nncan 8545	. . . . . 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 6090	. . . 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 9110	. . . 4 |- ( ph -> ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) e. CC )
90	1, 3, 32, 9, 64	isumcl 12170	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) e. CC )
91	89, 90	pncan2d 8629	. . 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 4151	1 |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   |-> cmpt 4187   dom cdm 4769   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   # cap 8899   / cdiv 8992   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   ^cexp 10953   abscabs 11741   ~~> cli 12022   sum_csu 12097

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  geoisum1  12264  geoisum1c  12265  trilpolemisumle  16992