Theorem geolim2 12229

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 2234	. . 3 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		geolim2.3	. . . 4 |- ( ph -> M e. NN0 )
3	2	nn0zd 9721	. . 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 9954	. . . . 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 11065	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( A ^ k ) e. CC )
10		eluzelz 9886	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
11	10	adantl 277	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ZZ )
12		0red 8293	. . . . . . . . 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 9723	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. RR )
15	11	zred 9723	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. RR )
16	2	nn0ge0d 9578	. . . . . . . . . 10 |- ( ph -> 0 <_ M )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ M )
18		eluzle 9889	. . . . . . . . . 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 8416	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ x )
21		elnn0z 9612	. . . . . . . 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 11065	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( A ^ x ) e. CC )
25		oveq2 6068	. . . . . . . 8 |- ( n = x -> ( A ^ n ) = ( A ^ x ) )
26		eqid 2234	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
27	25, 26	fvmptg 5760	. . . . . . 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 2311	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) e. CC )
30		oveq2 6068	. . . . . . . 8 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31	30, 26	fvmptg 5760	. . . . . . 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 2270	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( F ` k ) )
34		addcl 8270	. . . . . 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 10871	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) = seq M ( + , F ) )
37		seqex 10840	. . . . . 6 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
38		ax-1cn 8238	. . . . . . . 8 |- 1 e. CC
39		subcl 8491	. . . . . . . 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 8308	. . . . . . . 8 |- ( ph -> 1 e. CC )
42		1red 8307	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
43		geolim.2	. . . . . . . . . 10 |- ( ph -> ( abs ` A ) < 1 )
44	5, 42, 43	absltap 12226	. . . . . . . . 9 |- ( ph -> A # 1 )
45		apsym 8900	. . . . . . . . . 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 8938	. . . . . . 7 |- ( ph -> ( 1 - A ) # 0 )
49	40, 48	recclapd 9077	. . . . . 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 11065	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
53		oveq2 6068	. . . . . . . . 9 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
54	53, 26	fvmptg 5760	. . . . . . . 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 12228	. . . . . 6 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
57		breldmg 4969	. . . . . 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 1379	. . . . 5 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
59		nn0uz 9912	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
60		expcl 10948	. . . . . . . 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 2311	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
63	59, 2, 62	iserex 12055	. . . . 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 2312	. . 3 |- ( ph -> seq M ( + , F ) e. dom ~~> )
66	1, 3, 4, 9, 65	isumclim2 12139	. 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 11065	. . . . . . . 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 10948	. . . . . . . 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 12208	. . . . . 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 9611	. . . . . . 7 |- ( ph -> 0 e. ZZ )
75	59, 74, 70, 72, 56	isumclim 12138	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
76	73, 75	eqtr3d 2269	. . . . 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 12224	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) )
78	77	oveq1d 6075	. . . . 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 2269	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) = ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
80	79	oveq1d 6075	. . 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 11065	. . . . . 6 |- ( ph -> ( A ^ M ) e. CC )
82		subcl 8491	. . . . . 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 9126	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
85		nncan 8521	. . . . . 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 6075	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
88	84, 87	eqtr3d 2269	. . 3 |- ( ph -> ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
89	83, 40, 48	divclapd 9086	. . . 4 |- ( ph -> ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) e. CC )
90	1, 3, 32, 9, 64	isumcl 12142	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) e. CC )
91	89, 90	pncan2d 8605	. . 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 2276	. 2 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) = ( ( A ^ M ) / ( 1 - A ) ) )
93	66, 92	breqtrd 4141	1 |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   class class class wbr 4115   |-> cmpt 4177   dom cdm 4756   ` cfv 5359  (class class class)co 6060   CCcc 8143   0cc0 8145   1c1 8146   + caddc 8148   < clt 8326   <_ cle 8327   - cmin 8463   # cap 8875   / cdiv 8968   NN0cn0 9518   ZZcz 9599   ZZ>=cuz 9876   ...cfz 10366   seqcseq 10838   ^cexp 10929   abscabs 11713   ~~> cli 11994   sum_csu 12069

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263  ax-arch 8264  ax-caucvg 8265

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-isom 5368  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-irdg 6616  df-frec 6637  df-1o 6662  df-oadd 6666  df-er 6782  df-en 6991  df-dom 6992  df-fin 6993  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-n0 9519  df-z 9600  df-uz 9877  df-q 9975  df-rp 10010  df-fz 10367  df-fzo 10504  df-seqfrec 10839  df-exp 10930  df-ihash 11169  df-cj 11557  df-re 11558  df-im 11559  df-rsqrt 11714  df-abs 11715  df-clim 11995  df-sumdc 12070

This theorem is referenced by:  geoisum1  12236  geoisum1c  12237  trilpolemisumle  16963