Theorem geo2lim 12261

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 9937	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9650	. . 3 |- ( A e. CC -> 1 e. ZZ )
3		halfcn 9498	. . . . . . 7 |- ( 1 / 2 ) e. CC
4	3	a1i 9	. . . . . 6 |- ( A e. CC -> ( 1 / 2 ) e. CC )
5		halfre 9497	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
6		halfge0 9500	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
7		absid 11815	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
8	5, 6, 7	mp2an 430	. . . . . . . 8 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
9		halflt1 9501	. . . . . . . 8 |- ( 1 / 2 ) < 1
10	8, 9	eqbrtri 4146	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
11	10	a1i 9	. . . . . 6 |- ( A e. CC -> ( abs ` ( 1 / 2 ) ) < 1 )
12	4, 11	expcnv 12249	. . . . 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 9289	. . . . . . . 8 |- NN e. _V
16	15	mptex 5934	. . . . . . 7 |- ( k e. NN |-> ( A / ( 2 ^ k ) ) ) e. _V
17	14, 16	eqeltri 2311	. . . . . 6 |- F e. _V
18	17	a1i 9	. . . . 5 |- ( A e. CC -> F e. _V )
19		nnnn0 9549	. . . . . . . 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 11089	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) e. CC )
23		oveq2 6083	. . . . . . . 8 |- ( k = j -> ( ( 1 / 2 ) ^ k ) = ( ( 1 / 2 ) ^ j ) )
24		eqid 2238	. . . . . . . 8 |- ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) = ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) )
25	23, 24	fvmptg 5775	. . . . . . 7 |- ( ( j e. NN0 /\ ( ( 1 / 2 ) ^ j ) e. CC ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
26	20, 22, 25	syl2anc 415	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
27	26, 22	eqeltrd 2315	. . . . 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 9445	. . . . . . . . 9 |- 2 e. NN
30		nnexpcl 10967	. . . . . . . . 9 |- ( ( 2 e. NN /\ j e. NN0 ) -> ( 2 ^ j ) e. NN )
31	29, 20, 30	sylancr 418	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. NN )
32	31	nncnd 9297	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. CC )
33	31	nnap0d 9329	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) # 0 )
34	28, 32, 33	divrecapd 9113	. . . . . 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 9110	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) e. CC )
37		oveq2 6083	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
38	37	oveq2d 6091	. . . . . . . 8 |- ( k = j -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ j ) ) )
39	38, 14	fvmptg 5775	. . . . . . 7 |- ( ( j e. NN /\ ( A / ( 2 ^ j ) ) e. CC ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
40	35, 36, 39	syl2anc 415	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
41		2cn 9354	. . . . . . . . 9 |- 2 e. CC
42		2ap0 9376	. . . . . . . . 9 |- 2 # 0
43		nnz 9642	. . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
44	43	adantl 277	. . . . . . . . 9 |- ( ( A e. CC /\ j e. NN ) -> j e. ZZ )
45		exprecap 10995	. . . . . . . . 9 |- ( ( 2 e. CC /\ 2 # 0 /\ j e. ZZ ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
46	41, 42, 44, 45	mp3an12i 1382	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
47	26, 46	eqtrd 2271	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
48	47	oveq2d 6091	. . . . . 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 2281	. . . . 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 12075	. . . 4 |- ( A e. CC -> F ~~> ( A x. 0 ) )
51		mul01 8706	. . . 4 |- ( A e. CC -> ( A x. 0 ) = 0 )
52	50, 51	breqtrd 4151	. . 3 |- ( A e. CC -> F ~~> 0 )
53		seqex 10864	. . . 4 |- seq 1 ( + , F ) e. _V
54	53	a1i 9	. . 3 |- ( A e. CC -> seq 1 ( + , F ) e. _V )
55	40, 36	eqeltrd 2315	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) e. CC )
56	40	oveq2d 6091	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A - ( F ` j ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
57		geo2sum 12259	. . . . 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 9938	. . . . . . . 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 531	. . . . . . 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 9599	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN0 )
65	63, 64	expcld 11089	. . . . . . 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 9746	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. ZZ )
68	63, 66, 67	expap0d 11095	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( 2 ^ n ) # 0 )
69	62, 65, 68	divclapd 9110	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( A / ( 2 ^ n ) ) e. CC )
70		oveq2 6083	. . . . . . . 8 |- ( k = n -> ( 2 ^ k ) = ( 2 ^ n ) )
71	70	oveq2d 6091	. . . . . . 7 |- ( k = n -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ n ) ) )
72	71, 14	fvmptg 5775	. . . . . 6 |- ( ( n e. NN /\ ( A / ( 2 ^ n ) ) e. CC ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
73	61, 69, 72	syl2anc 415	. . . . 5 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
74	35, 1	eleqtrdi 2331	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> j e. ( ZZ>= ` 1 ) )
75	73, 74, 69	fsum3ser 12142	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( seq 1 ( + , F ) ` j ) )
76	56, 58, 75	3eqtr2rd 2278	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( seq 1 ( + , F ) ` j ) = ( A - ( F ` j ) ) )
77	1, 2, 52, 13, 54, 55, 76	climsubc2 12077	. 2 |- ( A e. CC -> seq 1 ( + , F ) ~~> ( A - 0 ) )
78		subid1 8536	. 2 |- ( A e. CC -> ( A - 0 ) = A )
79	77, 78	breqtrd 4151	1 |- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   # cap 8899   / cdiv 8992   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   ^cexp 10953   abscabs 11741   ~~> cli 12022   sum_csu 12097

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  trilpolemeq1  16994