Theorem geo2lim 11871

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 9691	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9406	. . 3 |- ( A e. CC -> 1 e. ZZ )
3		halfcn 9258	. . . . . . 7 |- ( 1 / 2 ) e. CC
4	3	a1i 9	. . . . . 6 |- ( A e. CC -> ( 1 / 2 ) e. CC )
5		halfre 9257	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
6		halfge0 9260	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
7		absid 11426	. . . . . . . . 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 9261	. . . . . . . 8 |- ( 1 / 2 ) < 1
10	8, 9	eqbrtri 4068	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
11	10	a1i 9	. . . . . 6 |- ( A e. CC -> ( abs ` ( 1 / 2 ) ) < 1 )
12	4, 11	expcnv 11859	. . . . 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 9049	. . . . . . . 8 |- NN e. _V
16	15	mptex 5817	. . . . . . 7 |- ( k e. NN |-> ( A / ( 2 ^ k ) ) ) e. _V
17	14, 16	eqeltri 2279	. . . . . 6 |- F e. _V
18	17	a1i 9	. . . . 5 |- ( A e. CC -> F e. _V )
19		nnnn0 9309	. . . . . . . 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 10825	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) e. CC )
23		oveq2 5959	. . . . . . . 8 |- ( k = j -> ( ( 1 / 2 ) ^ k ) = ( ( 1 / 2 ) ^ j ) )
24		eqid 2206	. . . . . . . 8 |- ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) = ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) )
25	23, 24	fvmptg 5662	. . . . . . 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 2283	. . . . 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 9205	. . . . . . . . 9 |- 2 e. NN
30		nnexpcl 10704	. . . . . . . . 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 9057	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. CC )
33	31	nnap0d 9089	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) # 0 )
34	28, 32, 33	divrecapd 8873	. . . . . 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 8870	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) e. CC )
37		oveq2 5959	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
38	37	oveq2d 5967	. . . . . . . 8 |- ( k = j -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ j ) ) )
39	38, 14	fvmptg 5662	. . . . . . 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 9114	. . . . . . . . 9 |- 2 e. CC
42		2ap0 9136	. . . . . . . . 9 |- 2 # 0
43		nnz 9398	. . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
44	43	adantl 277	. . . . . . . . 9 |- ( ( A e. CC /\ j e. NN ) -> j e. ZZ )
45		exprecap 10732	. . . . . . . . 9 |- ( ( 2 e. CC /\ 2 # 0 /\ j e. ZZ ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
46	41, 42, 44, 45	mp3an12i 1354	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
47	26, 46	eqtrd 2239	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
48	47	oveq2d 5967	. . . . . 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 2249	. . . . 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 11686	. . . 4 |- ( A e. CC -> F ~~> ( A x. 0 ) )
51		mul01 8468	. . . 4 |- ( A e. CC -> ( A x. 0 ) = 0 )
52	50, 51	breqtrd 4073	. . 3 |- ( A e. CC -> F ~~> 0 )
53		seqex 10601	. . . 4 |- seq 1 ( + , F ) e. _V
54	53	a1i 9	. . 3 |- ( A e. CC -> seq 1 ( + , F ) e. _V )
55	40, 36	eqeltrd 2283	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) e. CC )
56	40	oveq2d 5967	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A - ( F ` j ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
57		geo2sum 11869	. . . . 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 9692	. . . . . . . 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 9355	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN0 )
65	63, 64	expcld 10825	. . . . . . 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 9501	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. ZZ )
68	63, 66, 67	expap0d 10831	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( 2 ^ n ) # 0 )
69	62, 65, 68	divclapd 8870	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( A / ( 2 ^ n ) ) e. CC )
70		oveq2 5959	. . . . . . . 8 |- ( k = n -> ( 2 ^ k ) = ( 2 ^ n ) )
71	70	oveq2d 5967	. . . . . . 7 |- ( k = n -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ n ) ) )
72	71, 14	fvmptg 5662	. . . . . 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 2299	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> j e. ( ZZ>= ` 1 ) )
75	73, 74, 69	fsum3ser 11752	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( seq 1 ( + , F ) ` j ) )
76	56, 58, 75	3eqtr2rd 2246	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( seq 1 ( + , F ) ` j ) = ( A - ( F ` j ) ) )
77	1, 2, 52, 13, 54, 55, 76	climsubc2 11688	. 2 |- ( A e. CC -> seq 1 ( + , F ) ~~> ( A - 0 ) )
78		subid1 8299	. 2 |- ( A e. CC -> ( A - 0 ) = A )
79	77, 78	breqtrd 4073	1 |- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   _Vcvv 2773   class class class wbr 4047   |-> cmpt 4109   ` cfv 5276  (class class class)co 5951   CCcc 7930   RRcr 7931   0cc0 7932   1c1 7933   + caddc 7935   x. cmul 7937   < clt 8114   <_ cle 8115   - cmin 8250   # cap 8661   / cdiv 8752   NNcn 9043   2c2 9094   NN0cn0 9302   ZZcz 9379   ZZ>=cuz 9655   ...cfz 10137   seqcseq 10599   ^cexp 10690   abscabs 11352   ~~> cli 11633   sum_csu 11708

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 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

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 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-oadd 6513  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709

This theorem is referenced by:  trilpolemeq1  16053