Theorem geolim 11657

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 9630	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9332	. . 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 11650	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
6		ax-1cn 7967	. . . . . . 7 |- 1 e. CC
7		subcl 8220	. . . . . . 7 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
8	6, 3, 7	sylancr 414	. . . . . 6 |- ( ph -> ( 1 - A ) e. CC )
9		1cnd 8037	. . . . . . 7 |- ( ph -> 1 e. CC )
10		1red 8036	. . . . . . . . 9 |- ( ph -> 1 e. RR )
11	3, 10, 4	absltap 11655	. . . . . . . 8 |- ( ph -> A # 1 )
12		apsym 8627	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
13	3, 6, 12	sylancl 413	. . . . . . . 8 |- ( ph -> ( A # 1 <-> 1 # A ) )
14	11, 13	mpbid 147	. . . . . . 7 |- ( ph -> 1 # A )
15	9, 3, 14	subap0d 8665	. . . . . 6 |- ( ph -> ( 1 - A ) # 0 )
16	3, 8, 15	divclapd 8811	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
17		nn0ex 9249	. . . . . . 7 |- NN0 e. _V
18	17	mptex 5785	. . . . . 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 10747	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
23		oveq2 5927	. . . . . . . 8 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
24		eqid 2193	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
25	23, 24	fvmptg 5634	. . . . . . 7 |- ( ( j e. NN0 /\ ( A ^ j ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
26	20, 22, 25	syl2anc 411	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
27		expcl 10631	. . . . . . 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 2270	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
30		expp1 10620	. . . . . . . . . 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 8043	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ j ) x. A ) = ( A x. ( A ^ j ) ) )
33	31, 32	eqtrd 2226	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( A x. ( A ^ j ) ) )
34	33	oveq1d 5934	. . . . . . 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 8849	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
38	34, 37	eqtrd 2226	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
39		peano2nn0 9283	. . . . . . . . . 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 10747	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
42	41, 35, 36	divclapd 8811	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
43		oveq1 5926	. . . . . . . . . 10 |- ( n = j -> ( n + 1 ) = ( j + 1 ) )
44	43	oveq2d 5935	. . . . . . . . 9 |- ( n = j -> ( A ^ ( n + 1 ) ) = ( A ^ ( j + 1 ) ) )
45	44	oveq1d 5934	. . . . . . . 8 |- ( n = j -> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
46		eqid 2193	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) = ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) )
47	45, 46	fvmptg 5634	. . . . . . 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 411	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
49	26	oveq2d 5935	. . . . . 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 2236	. . . . 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 11477	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> ( ( A / ( 1 - A ) ) x. 0 ) )
52	16	mul01d 8414	. . . 4 |- ( ph -> ( ( A / ( 1 - A ) ) x. 0 ) = 0 )
53	51, 52	breqtrd 4056	. . 3 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> 0 )
54	8, 15	recclapd 8802	. . 3 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
55		seqex 10523	. . . 4 |- seq 0 ( + , F ) e. _V
56	55	a1i 9	. . 3 |- ( ph -> seq 0 ( + , F ) e. _V )
57		expcl 10631	. . . . . 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 8811	. . . 4 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
60	48, 59	eqeltrd 2270	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) e. CC )
61		nn0cn 9253	. . . . . . . 8 |- ( j e. NN0 -> j e. CC )
62	61	adantl 277	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. CC )
63		pncan 8227	. . . . . . 7 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
64	62, 6, 63	sylancl 413	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( j + 1 ) - 1 ) = j )
65	64	oveq2d 5935	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( 0 ... ( ( j + 1 ) - 1 ) ) = ( 0 ... j ) )
66	65	sumeq1d 11512	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = sum_ k e. ( 0 ... j ) ( A ^ k ) )
67		1cnd 8037	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> 1 e. CC )
68	67, 58, 35, 36	divsubdirapd 8851	. . . . 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 11653	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) )
71	48	oveq2d 5935	. . . . 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 2236	. . . 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 527	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ph )
74		elnn0uz 9633	. . . . . . . 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 411	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( A ^ k ) )
79	20, 1	eleqtrdi 2286	. . . . 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 10747	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. CC )
82	78, 79, 81	fsum3ser 11543	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( A ^ k ) = ( seq 0 ( + , F ) ` j ) )
83	66, 72, 82	3eqtr3rd 2235	. . 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 11479	. 2 |- ( ph -> seq 0 ( + , F ) ~~> ( ( 1 / ( 1 - A ) ) - 0 ) )
85	54	subid1d 8321	. 2 |- ( ph -> ( ( 1 / ( 1 - A ) ) - 0 ) = ( 1 / ( 1 - A ) ) )
86	84, 85	breqtrd 4056	1 |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   |-> cmpt 4091   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   < clt 8056   - cmin 8192   # cap 8602   / cdiv 8693   NN0cn0 9243   ZZ>=cuz 9595   ...cfz 10077   seqcseq 10521   ^cexp 10612   abscabs 11144   ~~> cli 11424   sum_csu 11499

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500

This theorem is referenced by:  geolim2  11658  georeclim  11659  geoisum  11663  eflegeo  11847