Theorem geolim2 11313

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 2140	. . 3 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		geolim2.3	. . . 4 |- ( ph -> M e. NN0 )
3	2	nn0zd 9195	. . 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 274	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
7		eluznn0 9420	. . . . 5 |- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
8	2, 7	sylan 281	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
9	6, 8	expcld 10455	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( A ^ k ) e. CC )
10		eluzelz 9359	. . . . . . . . 9 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
11	10	adantl 275	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ZZ )
12		0red 7791	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 e. RR )
13	3	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. ZZ )
14	13	zred 9197	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M e. RR )
15	11	zred 9197	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. RR )
16	2	nn0ge0d 9057	. . . . . . . . . 10 |- ( ph -> 0 <_ M )
17	16	adantr 274	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ M )
18		eluzle 9362	. . . . . . . . . 10 |- ( x e. ( ZZ>= ` M ) -> M <_ x )
19	18	adantl 275	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> M <_ x )
20	12, 14, 15, 17, 19	letrd 7910	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 <_ x )
21		elnn0z 9091	. . . . . . . 8 |- ( x e. NN0 <-> ( x e. ZZ /\ 0 <_ x ) )
22	11, 20, 21	sylanbrc 414	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. NN0 )
23	5	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A e. CC )
24	23, 22	expcld 10455	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( A ^ x ) e. CC )
25		oveq2 5790	. . . . . . . 8 |- ( n = x -> ( A ^ n ) = ( A ^ x ) )
26		eqid 2140	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
27	25, 26	fvmptg 5505	. . . . . . 7 |- ( ( x e. NN0 /\ ( A ^ x ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) = ( A ^ x ) )
28	22, 24, 27	syl2anc 409	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) = ( A ^ x ) )
29	28, 24	eqeltrd 2217	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` x ) e. CC )
30		oveq2 5790	. . . . . . . 8 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31	30, 26	fvmptg 5505	. . . . . . 7 |- ( ( k e. NN0 /\ ( A ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
32	8, 9, 31	syl2anc 409	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
33	32, 4	eqtr4d 2176	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( F ` k ) )
34		addcl 7769	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
35	34	adantl 275	. . . . 5 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
36	3, 29, 33, 35	seq3feq 10276	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) = seq M ( + , F ) )
37		seqex 10251	. . . . . 6 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
38		ax-1cn 7737	. . . . . . . 8 |- 1 e. CC
39		subcl 7985	. . . . . . . 8 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
40	38, 5, 39	sylancr 411	. . . . . . 7 |- ( ph -> ( 1 - A ) e. CC )
41		1cnd 7806	. . . . . . . 8 |- ( ph -> 1 e. CC )
42		1red 7805	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
43		geolim.2	. . . . . . . . . 10 |- ( ph -> ( abs ` A ) < 1 )
44	5, 42, 43	absltap 11310	. . . . . . . . 9 |- ( ph -> A # 1 )
45		apsym 8392	. . . . . . . . . 10 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
46	5, 41, 45	syl2anc 409	. . . . . . . . 9 |- ( ph -> ( A # 1 <-> 1 # A ) )
47	44, 46	mpbid 146	. . . . . . . 8 |- ( ph -> 1 # A )
48	41, 5, 47	subap0d 8430	. . . . . . 7 |- ( ph -> ( 1 - A ) # 0 )
49	40, 48	recclapd 8565	. . . . . 6 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
50		simpr 109	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> j e. NN0 )
51	5	adantr 274	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> A e. CC )
52	51, 50	expcld 10455	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
53		oveq2 5790	. . . . . . . . 9 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
54	53, 26	fvmptg 5505	. . . . . . . 8 |- ( ( j e. NN0 /\ ( A ^ j ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
55	50, 52, 54	syl2anc 409	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
56	5, 43, 55	geolim 11312	. . . . . 6 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
57		breldmg 4753	. . . . . 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 1321	. . . . 5 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
59		nn0uz 9384	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
60		expcl 10342	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ j ) e. CC )
61	5, 60	sylan 281	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
62	55, 61	eqeltrd 2217	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
63	59, 2, 62	iserex 11140	. . . . 5 |- ( ph -> ( seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> <-> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> ) )
64	58, 63	mpbid 146	. . . 4 |- ( ph -> seq M ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
65	36, 64	eqeltrrd 2218	. . 3 |- ( ph -> seq M ( + , F ) e. dom ~~> )
66	1, 3, 4, 9, 65	isumclim2 11223	. 2 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) )
67		simpr 109	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
68	5	adantr 274	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> A e. CC )
69	68, 67	expcld 10455	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
70	67, 69, 31	syl2anc 409	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
71		expcl 10342	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
72	5, 71	sylan 281	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
73	59, 1, 2, 70, 72, 58	isumsplit 11292	. . . . . 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 9090	. . . . . . 7 |- ( ph -> 0 e. ZZ )
75	59, 74, 70, 72, 56	isumclim 11222	. . . . . 6 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
76	73, 75	eqtr3d 2175	. . . . 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 11308	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( M - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) )
78	77	oveq1d 5797	. . . . 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 2175	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) = ( ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) + sum_ k e. ( ZZ>= ` M ) ( A ^ k ) ) )
80	79	oveq1d 5797	. . 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 10455	. . . . . 6 |- ( ph -> ( A ^ M ) e. CC )
82		subcl 7985	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( A ^ M ) ) e. CC )
83	38, 81, 82	sylancr 411	. . . . 5 |- ( ph -> ( 1 - ( A ^ M ) ) e. CC )
84	41, 83, 40, 48	divsubdirapd 8614	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) )
85		nncan 8015	. . . . . 6 |- ( ( 1 e. CC /\ ( A ^ M ) e. CC ) -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
86	38, 81, 85	sylancr 411	. . . . 5 |- ( ph -> ( 1 - ( 1 - ( A ^ M ) ) ) = ( A ^ M ) )
87	86	oveq1d 5797	. . . 4 |- ( ph -> ( ( 1 - ( 1 - ( A ^ M ) ) ) / ( 1 - A ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
88	84, 87	eqtr3d 2175	. . 3 |- ( ph -> ( ( 1 / ( 1 - A ) ) - ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) ) = ( ( A ^ M ) / ( 1 - A ) ) )
89	83, 40, 48	divclapd 8574	. . . 4 |- ( ph -> ( ( 1 - ( A ^ M ) ) / ( 1 - A ) ) e. CC )
90	1, 3, 32, 9, 64	isumcl 11226	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) e. CC )
91	89, 90	pncan2d 8099	. . 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 2182	. 2 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( A ^ k ) = ( ( A ^ M ) / ( 1 - A ) ) )
93	66, 92	breqtrd 3962	1 |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   _Vcvv 2689   class class class wbr 3937   |-> cmpt 3997   dom cdm 4547   ` cfv 5131  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   < clt 7824   <_ cle 7825   - cmin 7957   # cap 8367   / cdiv 8456   NN0cn0 9001   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821   seqcseq 10249   ^cexp 10323   abscabs 10801   ~~> cli 11079   sum_csu 11154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by:  geoisum1  11320  geoisum1c  11321  trilpolemisumle  13406