Theorem geolim 12303

Description: The partial sums in the infinite series 1 + A ^ 1 + A ^ 2... converge to ( 1 / ( 1 - A ) ). (Contributed by NM, 15-May-2006.)

Hypotheses

Ref	Expression
geolim.1	|- ( ph -> A e. CC )
geolim.2	|- ( ph -> ( abs ` A ) < 1 )
geolim.3	|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( A ^ k ) )

Assertion

Ref	Expression
geolim	|- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )

Distinct variable groups:   A, k   k, F   ph, k

Proof of Theorem geolim

Dummy variables j n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 9973	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9667	. . 3 |- ( ph -> 0 e. ZZ )
3		geolim.1	. . . . . 6 |- ( ph -> A e. CC )
4		geolim.2	. . . . . 6 |- ( ph -> ( abs ` A ) < 1 )
5	3, 4	expcnv 12296	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
6		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
7		subcl 8532	. . . . . . 7 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
8	6, 3, 7	sylancr 418	. . . . . 6 |- ( ph -> ( 1 - A ) e. CC )
9		1cnd 8348	. . . . . . 7 |- ( ph -> 1 e. CC )
10		1red 8347	. . . . . . . . 9 |- ( ph -> 1 e. RR )
11	3, 10, 4	absltap 12301	. . . . . . . 8 |- ( ph -> A # 1 )
12		apsym 8942	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
13	3, 6, 12	sylancl 417	. . . . . . . 8 |- ( ph -> ( A # 1 <-> 1 # A ) )
14	11, 13	mpbid 147	. . . . . . 7 |- ( ph -> 1 # A )
15	9, 3, 14	subap0d 8980	. . . . . 6 |- ( ph -> ( 1 - A ) # 0 )
16	3, 8, 15	divclapd 9128	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
17		nn0ex 9580	. . . . . . 7 |- NN0 e. _V
18	17	mptex 5946	. . . . . 6 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) e. _V
19	18	a1i 9	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) e. _V )
20		simpr 110	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. NN0 )
21	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> A e. CC )
22	21, 20	expcld 11132	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
23		oveq2 6096	. . . . . . . 8 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
24		eqid 2238	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
25	23, 24	fvmptg 5784	. . . . . . 7 |- ( ( j e. NN0 /\ ( A ^ j ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
26	20, 22, 25	syl2anc 415	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
27		expcl 11015	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ j ) e. CC )
28	3, 27	sylan 283	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
29	26, 28	eqeltrd 2315	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
30		expp1 11004	. . . . . . . . . 10 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( ( A ^ j ) x. A ) )
31	3, 30	sylan 283	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( ( A ^ j ) x. A ) )
32	28, 21	mulcomd 8353	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ j ) x. A ) = ( A x. ( A ^ j ) ) )
33	31, 32	eqtrd 2271	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( A x. ( A ^ j ) ) )
34	33	oveq1d 6103	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) )
35	8	adantr 276	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( 1 - A ) e. CC )
36	15	adantr 276	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( 1 - A ) # 0 )
37	21, 28, 35, 36	div23apd 9166	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
38	34, 37	eqtrd 2271	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
39		peano2nn0 9614	. . . . . . . . . 10 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
40	39	adantl 277	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( j + 1 ) e. NN0 )
41	21, 40	expcld 11132	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
42	41, 35, 36	divclapd 9128	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
43		oveq1 6095	. . . . . . . . . 10 |- ( n = j -> ( n + 1 ) = ( j + 1 ) )
44	43	oveq2d 6104	. . . . . . . . 9 |- ( n = j -> ( A ^ ( n + 1 ) ) = ( A ^ ( j + 1 ) ) )
45	44	oveq1d 6103	. . . . . . . 8 |- ( n = j -> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
46		eqid 2238	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) = ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) )
47	45, 46	fvmptg 5784	. . . . . . 7 |- ( ( j e. NN0 /\ ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
48	20, 42, 47	syl2anc 415	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
49	26	oveq2d 6104	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A / ( 1 - A ) ) x. ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
50	38, 48, 49	3eqtr4d 2281	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A / ( 1 - A ) ) x. ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) ) )
51	1, 2, 5, 16, 19, 29, 50	climmulc2 12122	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> ( ( A / ( 1 - A ) ) x. 0 ) )
52	16	mul01d 8727	. . . 4 |- ( ph -> ( ( A / ( 1 - A ) ) x. 0 ) = 0 )
53	51, 52	breqtrd 4157	. . 3 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> 0 )
54	8, 15	recclapd 9119	. . 3 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
55		seqex 10907	. . . 4 |- seq 0 ( + , F ) e. _V
56	55	a1i 9	. . 3 |- ( ph -> seq 0 ( + , F ) e. _V )
57		expcl 11015	. . . . . 6 |- ( ( A e. CC /\ ( j + 1 ) e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
58	3, 39, 57	syl2an 289	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
59	58, 35, 36	divclapd 9128	. . . 4 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
60	48, 59	eqeltrd 2315	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) e. CC )
61		nn0cn 9584	. . . . . . . 8 |- ( j e. NN0 -> j e. CC )
62	61	adantl 277	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. CC )
63		pncan 8539	. . . . . . 7 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
64	62, 6, 63	sylancl 417	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( j + 1 ) - 1 ) = j )
65	64	oveq2d 6104	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( 0 ... ( ( j + 1 ) - 1 ) ) = ( 0 ... j ) )
66	65	sumeq1d 12157	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = sum_ k e. ( 0 ... j ) ( A ^ k ) )
67		1cnd 8348	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> 1 e. CC )
68	67, 58, 35, 36	divsubdirapd 9168	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) ) )
69	11	adantr 276	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> A # 1 )
70	21, 69, 40	geoserap 12299	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) )
71	48	oveq2d 6104	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( 1 / ( 1 - A ) ) - ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) ) = ( ( 1 / ( 1 - A ) ) - ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) ) )
72	68, 70, 71	3eqtr4d 2281	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 / ( 1 - A ) ) - ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) ) )
73		simpll 531	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ph )
74		elnn0uz 9976	. . . . . . . 8 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
75	74	biimpri 133	. . . . . . 7 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
76	75	adantl 277	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
77		geolim.3	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( A ^ k ) )
78	73, 76, 77	syl2anc 415	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( A ^ k ) )
79	20, 1	eleqtrdi 2331	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> j e. ( ZZ>= ` 0 ) )
80	21	adantr 276	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> A e. CC )
81	80, 76	expcld 11132	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. CC )
82	78, 79, 81	fsum3ser 12189	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( A ^ k ) = ( seq 0 ( + , F ) ` j ) )
83	66, 72, 82	3eqtr3rd 2280	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( seq 0 ( + , F ) ` j ) = ( ( 1 / ( 1 - A ) ) - ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) ) )
84	1, 2, 53, 54, 56, 60, 83	climsubc2 12124	. 2 |- ( ph -> seq 0 ( + , F ) ~~> ( ( 1 / ( 1 - A ) ) - 0 ) )
85	54	subid1d 8633	. 2 |- ( ph -> ( ( 1 / ( 1 - A ) ) - 0 ) = ( 1 / ( 1 - A ) ) )
86	84, 85	breqtrd 4157	1 |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 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 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   - cmin 8504   # cap 8917   / cdiv 9010   NN0cn0 9574   ZZ>=cuz 9937   ...cfz 10428   seqcseq 10905   ^cexp 10996   abscabs 11785   ~~> cli 12069   sum_csu 12144

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145

This theorem is used by:  geolim2  12304  georeclim  12305  geoisum  12309  eflegeo  12493