Theorem geolim 11897

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 9703	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9404	. . 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 11890	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
6		ax-1cn 8038	. . . . . . 7 |- 1 e. CC
7		subcl 8291	. . . . . . 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 8108	. . . . . . 7 |- ( ph -> 1 e. CC )
10		1red 8107	. . . . . . . . 9 |- ( ph -> 1 e. RR )
11	3, 10, 4	absltap 11895	. . . . . . . 8 |- ( ph -> A # 1 )
12		apsym 8699	. . . . . . . . 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 8737	. . . . . 6 |- ( ph -> ( 1 - A ) # 0 )
16	3, 8, 15	divclapd 8883	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
17		nn0ex 9321	. . . . . . 7 |- NN0 e. _V
18	17	mptex 5823	. . . . . 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 10840	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
23		oveq2 5965	. . . . . . . 8 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
24		eqid 2206	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
25	23, 24	fvmptg 5668	. . . . . . 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 10724	. . . . . . 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 2283	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
30		expp1 10713	. . . . . . . . . 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 8114	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ j ) x. A ) = ( A x. ( A ^ j ) ) )
33	31, 32	eqtrd 2239	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( A x. ( A ^ j ) ) )
34	33	oveq1d 5972	. . . . . . 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 8921	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
38	34, 37	eqtrd 2239	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
39		peano2nn0 9355	. . . . . . . . . 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 10840	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
42	41, 35, 36	divclapd 8883	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
43		oveq1 5964	. . . . . . . . . 10 |- ( n = j -> ( n + 1 ) = ( j + 1 ) )
44	43	oveq2d 5973	. . . . . . . . 9 |- ( n = j -> ( A ^ ( n + 1 ) ) = ( A ^ ( j + 1 ) ) )
45	44	oveq1d 5972	. . . . . . . 8 |- ( n = j -> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
46		eqid 2206	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) = ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) )
47	45, 46	fvmptg 5668	. . . . . . 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 5973	. . . . . 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 2249	. . . . 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 11717	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> ( ( A / ( 1 - A ) ) x. 0 ) )
52	16	mul01d 8485	. . . 4 |- ( ph -> ( ( A / ( 1 - A ) ) x. 0 ) = 0 )
53	51, 52	breqtrd 4077	. . 3 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> 0 )
54	8, 15	recclapd 8874	. . 3 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
55		seqex 10616	. . . 4 |- seq 0 ( + , F ) e. _V
56	55	a1i 9	. . 3 |- ( ph -> seq 0 ( + , F ) e. _V )
57		expcl 10724	. . . . . 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 8883	. . . 4 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
60	48, 59	eqeltrd 2283	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) e. CC )
61		nn0cn 9325	. . . . . . . 8 |- ( j e. NN0 -> j e. CC )
62	61	adantl 277	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. CC )
63		pncan 8298	. . . . . . 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 5973	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( 0 ... ( ( j + 1 ) - 1 ) ) = ( 0 ... j ) )
66	65	sumeq1d 11752	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = sum_ k e. ( 0 ... j ) ( A ^ k ) )
67		1cnd 8108	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> 1 e. CC )
68	67, 58, 35, 36	divsubdirapd 8923	. . . . 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 11893	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) )
71	48	oveq2d 5973	. . . . 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 2249	. . . 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 9706	. . . . . . . 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 2299	. . . . 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 10840	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. CC )
82	78, 79, 81	fsum3ser 11783	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( A ^ k ) = ( seq 0 ( + , F ) ` j ) )
83	66, 72, 82	3eqtr3rd 2248	. . 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 11719	. 2 |- ( ph -> seq 0 ( + , F ) ~~> ( ( 1 / ( 1 - A ) ) - 0 ) )
85	54	subid1d 8392	. 2 |- ( ph -> ( ( 1 / ( 1 - A ) ) - 0 ) = ( 1 / ( 1 - A ) ) )
86	84, 85	breqtrd 4077	1 |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   _Vcvv 2773   class class class wbr 4051   |-> cmpt 4113   ` cfv 5280  (class class class)co 5957   CCcc 7943   0cc0 7945   1c1 7946   + caddc 7948   x. cmul 7950   < clt 8127   - cmin 8263   # cap 8674   / cdiv 8765   NN0cn0 9315   ZZ>=cuz 9668   ...cfz 10150   seqcseq 10614   ^cexp 10705   abscabs 11383   ~~> cli 11664   sum_csu 11739

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-frec 6490  df-1o 6515  df-oadd 6519  df-er 6633  df-en 6841  df-dom 6842  df-fin 6843  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-seqfrec 10615  df-exp 10706  df-ihash 10943  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-clim 11665  df-sumdc 11740

This theorem is referenced by:  geolim2  11898  georeclim  11899  geoisum  11903  eflegeo  12087