Theorem geolim 11452

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 9500	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9203	. . 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 11445	. . . . 5 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
6		ax-1cn 7846	. . . . . . 7 |- 1 e. CC
7		subcl 8097	. . . . . . 7 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
8	6, 3, 7	sylancr 411	. . . . . 6 |- ( ph -> ( 1 - A ) e. CC )
9		1cnd 7915	. . . . . . 7 |- ( ph -> 1 e. CC )
10		1red 7914	. . . . . . . . 9 |- ( ph -> 1 e. RR )
11	3, 10, 4	absltap 11450	. . . . . . . 8 |- ( ph -> A # 1 )
12		apsym 8504	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
13	3, 6, 12	sylancl 410	. . . . . . . 8 |- ( ph -> ( A # 1 <-> 1 # A ) )
14	11, 13	mpbid 146	. . . . . . 7 |- ( ph -> 1 # A )
15	9, 3, 14	subap0d 8542	. . . . . 6 |- ( ph -> ( 1 - A ) # 0 )
16	3, 8, 15	divclapd 8686	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
17		nn0ex 9120	. . . . . . 7 |- NN0 e. _V
18	17	mptex 5711	. . . . . 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 109	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. NN0 )
21	3	adantr 274	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> A e. CC )
22	21, 20	expcld 10588	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
23		oveq2 5850	. . . . . . . 8 |- ( n = j -> ( A ^ n ) = ( A ^ j ) )
24		eqid 2165	. . . . . . . 8 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
25	23, 24	fvmptg 5562	. . . . . . 7 |- ( ( j e. NN0 /\ ( A ^ j ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
26	20, 22, 25	syl2anc 409	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) = ( A ^ j ) )
27		expcl 10473	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ j ) e. CC )
28	3, 27	sylan 281	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( A ^ j ) e. CC )
29	26, 28	eqeltrd 2243	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` j ) e. CC )
30		expp1 10462	. . . . . . . . . 10 |- ( ( A e. CC /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( ( A ^ j ) x. A ) )
31	3, 30	sylan 281	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( ( A ^ j ) x. A ) )
32	28, 21	mulcomd 7920	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ j ) x. A ) = ( A x. ( A ^ j ) ) )
33	31, 32	eqtrd 2198	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) = ( A x. ( A ^ j ) ) )
34	33	oveq1d 5857	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) )
35	8	adantr 274	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( 1 - A ) e. CC )
36	15	adantr 274	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( 1 - A ) # 0 )
37	21, 28, 35, 36	div23apd 8724	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A x. ( A ^ j ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
38	34, 37	eqtrd 2198	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) = ( ( A / ( 1 - A ) ) x. ( A ^ j ) ) )
39		peano2nn0 9154	. . . . . . . . . 10 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
40	39	adantl 275	. . . . . . . . 9 |- ( ( ph /\ j e. NN0 ) -> ( j + 1 ) e. NN0 )
41	21, 40	expcld 10588	. . . . . . . 8 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
42	41, 35, 36	divclapd 8686	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
43		oveq1 5849	. . . . . . . . . 10 |- ( n = j -> ( n + 1 ) = ( j + 1 ) )
44	43	oveq2d 5858	. . . . . . . . 9 |- ( n = j -> ( A ^ ( n + 1 ) ) = ( A ^ ( j + 1 ) ) )
45	44	oveq1d 5857	. . . . . . . 8 |- ( n = j -> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
46		eqid 2165	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) = ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) )
47	45, 46	fvmptg 5562	. . . . . . 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 409	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) = ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) )
49	26	oveq2d 5858	. . . . . 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 2208	. . . . 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 11272	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> ( ( A / ( 1 - A ) ) x. 0 ) )
52	16	mul01d 8291	. . . 4 |- ( ph -> ( ( A / ( 1 - A ) ) x. 0 ) = 0 )
53	51, 52	breqtrd 4008	. . 3 |- ( ph -> ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ~~> 0 )
54	8, 15	recclapd 8677	. . 3 |- ( ph -> ( 1 / ( 1 - A ) ) e. CC )
55		seqex 10382	. . . 4 |- seq 0 ( + , F ) e. _V
56	55	a1i 9	. . 3 |- ( ph -> seq 0 ( + , F ) e. _V )
57		expcl 10473	. . . . . 6 |- ( ( A e. CC /\ ( j + 1 ) e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
58	3, 39, 57	syl2an 287	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( A ^ ( j + 1 ) ) e. CC )
59	58, 35, 36	divclapd 8686	. . . 4 |- ( ( ph /\ j e. NN0 ) -> ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) e. CC )
60	48, 59	eqeltrd 2243	. . 3 |- ( ( ph /\ j e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ ( n + 1 ) ) / ( 1 - A ) ) ) ` j ) e. CC )
61		nn0cn 9124	. . . . . . . 8 |- ( j e. NN0 -> j e. CC )
62	61	adantl 275	. . . . . . 7 |- ( ( ph /\ j e. NN0 ) -> j e. CC )
63		pncan 8104	. . . . . . 7 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
64	62, 6, 63	sylancl 410	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> ( ( j + 1 ) - 1 ) = j )
65	64	oveq2d 5858	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( 0 ... ( ( j + 1 ) - 1 ) ) = ( 0 ... j ) )
66	65	sumeq1d 11307	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = sum_ k e. ( 0 ... j ) ( A ^ k ) )
67		1cnd 7915	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> 1 e. CC )
68	67, 58, 35, 36	divsubdirapd 8726	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) = ( ( 1 / ( 1 - A ) ) - ( ( A ^ ( j + 1 ) ) / ( 1 - A ) ) ) )
69	11	adantr 274	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> A # 1 )
70	21, 69, 40	geoserap 11448	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... ( ( j + 1 ) - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ ( j + 1 ) ) ) / ( 1 - A ) ) )
71	48	oveq2d 5858	. . . . 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 2208	. . . 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 519	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ph )
74		elnn0uz 9503	. . . . . . . 8 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
75	74	biimpri 132	. . . . . . 7 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
76	75	adantl 275	. . . . . 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 409	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( A ^ k ) )
79	20, 1	eleqtrdi 2259	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> j e. ( ZZ>= ` 0 ) )
80	21	adantr 274	. . . . . 6 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> A e. CC )
81	80, 76	expcld 10588	. . . . 5 |- ( ( ( ph /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. CC )
82	78, 79, 81	fsum3ser 11338	. . . 4 |- ( ( ph /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( A ^ k ) = ( seq 0 ( + , F ) ` j ) )
83	66, 72, 82	3eqtr3rd 2207	. . 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 11274	. 2 |- ( ph -> seq 0 ( + , F ) ~~> ( ( 1 / ( 1 - A ) ) - 0 ) )
85	54	subid1d 8198	. 2 |- ( ph -> ( ( 1 / ( 1 - A ) ) - 0 ) = ( 1 / ( 1 - A ) ) )
86	84, 85	breqtrd 4008	1 |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   |-> cmpt 4043   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   - cmin 8069   # cap 8479   / cdiv 8568   NN0cn0 9114   ZZ>=cuz 9466   ...cfz 9944   seqcseq 10380   ^cexp 10454   abscabs 10939   ~~> cli 11219   sum_csu 11294

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by:  geolim2  11453  georeclim  11454  geoisum  11458  eflegeo  11642