Theorem geo2lim 11681

Description: The value of the infinite geometric series 2 ^ -u 1 + 2 ^ -u 2 +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)

Hypothesis

Ref	Expression
geo2lim.1	|- F = ( k e. NN |-> ( A / ( 2 ^ k ) ) )

Assertion

Ref	Expression
geo2lim	|- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Distinct variable group:   A, k

Allowed substitution hint:   F( k)

Proof of Theorem geo2lim

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

Step	Hyp	Ref	Expression
1		nnuz 9637	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9353	. . 3 |- ( A e. CC -> 1 e. ZZ )
3		halfcn 9205	. . . . . . 7 |- ( 1 / 2 ) e. CC
4	3	a1i 9	. . . . . 6 |- ( A e. CC -> ( 1 / 2 ) e. CC )
5		halfre 9204	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
6		halfge0 9207	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
7		absid 11236	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
8	5, 6, 7	mp2an 426	. . . . . . . 8 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
9		halflt1 9208	. . . . . . . 8 |- ( 1 / 2 ) < 1
10	8, 9	eqbrtri 4054	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
11	10	a1i 9	. . . . . 6 |- ( A e. CC -> ( abs ` ( 1 / 2 ) ) < 1 )
12	4, 11	expcnv 11669	. . . . 5 |- ( A e. CC -> ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ~~> 0 )
13		id 19	. . . . 5 |- ( A e. CC -> A e. CC )
14		geo2lim.1	. . . . . . 7 |- F = ( k e. NN |-> ( A / ( 2 ^ k ) ) )
15		nnex 8996	. . . . . . . 8 |- NN e. _V
16	15	mptex 5788	. . . . . . 7 |- ( k e. NN |-> ( A / ( 2 ^ k ) ) ) e. _V
17	14, 16	eqeltri 2269	. . . . . 6 |- F e. _V
18	17	a1i 9	. . . . 5 |- ( A e. CC -> F e. _V )
19		nnnn0 9256	. . . . . . . 8 |- ( j e. NN -> j e. NN0 )
20	19	adantl 277	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> j e. NN0 )
21	3	a1i 9	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( 1 / 2 ) e. CC )
22	21, 20	expcld 10765	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) e. CC )
23		oveq2 5930	. . . . . . . 8 |- ( k = j -> ( ( 1 / 2 ) ^ k ) = ( ( 1 / 2 ) ^ j ) )
24		eqid 2196	. . . . . . . 8 |- ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) = ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) )
25	23, 24	fvmptg 5637	. . . . . . 7 |- ( ( j e. NN0 /\ ( ( 1 / 2 ) ^ j ) e. CC ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
26	20, 22, 25	syl2anc 411	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
27	26, 22	eqeltrd 2273	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) e. CC )
28		simpl 109	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> A e. CC )
29		2nn 9152	. . . . . . . . 9 |- 2 e. NN
30		nnexpcl 10644	. . . . . . . . 9 |- ( ( 2 e. NN /\ j e. NN0 ) -> ( 2 ^ j ) e. NN )
31	29, 20, 30	sylancr 414	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. NN )
32	31	nncnd 9004	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. CC )
33	31	nnap0d 9036	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) # 0 )
34	28, 32, 33	divrecapd 8820	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) = ( A x. ( 1 / ( 2 ^ j ) ) ) )
35		simpr 110	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> j e. NN )
36	28, 32, 33	divclapd 8817	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) e. CC )
37		oveq2 5930	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
38	37	oveq2d 5938	. . . . . . . 8 |- ( k = j -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ j ) ) )
39	38, 14	fvmptg 5637	. . . . . . 7 |- ( ( j e. NN /\ ( A / ( 2 ^ j ) ) e. CC ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
40	35, 36, 39	syl2anc 411	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
41		2cn 9061	. . . . . . . . 9 |- 2 e. CC
42		2ap0 9083	. . . . . . . . 9 |- 2 # 0
43		nnz 9345	. . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
44	43	adantl 277	. . . . . . . . 9 |- ( ( A e. CC /\ j e. NN ) -> j e. ZZ )
45		exprecap 10672	. . . . . . . . 9 |- ( ( 2 e. CC /\ 2 # 0 /\ j e. ZZ ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
46	41, 42, 44, 45	mp3an12i 1352	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
47	26, 46	eqtrd 2229	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
48	47	oveq2d 5938	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( A x. ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) ) = ( A x. ( 1 / ( 2 ^ j ) ) ) )
49	34, 40, 48	3eqtr4d 2239	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) = ( A x. ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) ) )
50	1, 2, 12, 13, 18, 27, 49	climmulc2 11496	. . . 4 |- ( A e. CC -> F ~~> ( A x. 0 ) )
51		mul01 8415	. . . 4 |- ( A e. CC -> ( A x. 0 ) = 0 )
52	50, 51	breqtrd 4059	. . 3 |- ( A e. CC -> F ~~> 0 )
53		seqex 10541	. . . 4 |- seq 1 ( + , F ) e. _V
54	53	a1i 9	. . 3 |- ( A e. CC -> seq 1 ( + , F ) e. _V )
55	40, 36	eqeltrd 2273	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) e. CC )
56	40	oveq2d 5938	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A - ( F ` j ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
57		geo2sum 11679	. . . . 5 |- ( ( j e. NN /\ A e. CC ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
58	57	ancoms 268	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
59		elnnuz 9638	. . . . . . . 8 |- ( n e. NN <-> n e. ( ZZ>= ` 1 ) )
60	59	biimpri 133	. . . . . . 7 |- ( n e. ( ZZ>= ` 1 ) -> n e. NN )
61	60	adantl 277	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN )
62		simpll 527	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> A e. CC )
63	41	a1i 9	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> 2 e. CC )
64	61	nnnn0d 9302	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN0 )
65	63, 64	expcld 10765	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( 2 ^ n ) e. CC )
66	42	a1i 9	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> 2 # 0 )
67	61	nnzd 9447	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. ZZ )
68	63, 66, 67	expap0d 10771	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( 2 ^ n ) # 0 )
69	62, 65, 68	divclapd 8817	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( A / ( 2 ^ n ) ) e. CC )
70		oveq2 5930	. . . . . . . 8 |- ( k = n -> ( 2 ^ k ) = ( 2 ^ n ) )
71	70	oveq2d 5938	. . . . . . 7 |- ( k = n -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ n ) ) )
72	71, 14	fvmptg 5637	. . . . . 6 |- ( ( n e. NN /\ ( A / ( 2 ^ n ) ) e. CC ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
73	61, 69, 72	syl2anc 411	. . . . 5 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
74	35, 1	eleqtrdi 2289	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> j e. ( ZZ>= ` 1 ) )
75	73, 74, 69	fsum3ser 11562	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( seq 1 ( + , F ) ` j ) )
76	56, 58, 75	3eqtr2rd 2236	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( seq 1 ( + , F ) ` j ) = ( A - ( F ` j ) ) )
77	1, 2, 52, 13, 54, 55, 76	climsubc2 11498	. 2 |- ( A e. CC -> seq 1 ( + , F ) ~~> ( A - 0 ) )
78		subid1 8246	. 2 |- ( A e. CC -> ( A - 0 ) = A )
79	77, 78	breqtrd 4059	1 |- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   <_ cle 8062   - cmin 8197   # cap 8608   / cdiv 8699   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539   ^cexp 10630   abscabs 11162   ~~> cli 11443   sum_csu 11518

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519

This theorem is referenced by:  trilpolemeq1  15684