Theorem geo2lim 14606

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 11723	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 11408	. . 3 |- ( A e. CC -> 1 e. ZZ )
3		halfcn 11247	. . . . . . 7 |- ( 1 / 2 ) e. CC
4	3	a1i 11	. . . . . 6 |- ( A e. CC -> ( 1 / 2 ) e. CC )
5		halfre 11246	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
6		0re 10040	. . . . . . . . . 10 |- 0 e. RR
7		halfgt0 11248	. . . . . . . . . 10 |- 0 < ( 1 / 2 )
8	6, 5, 7	ltleii 10160	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
9		absid 14036	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
10	5, 8, 9	mp2an 708	. . . . . . . 8 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
11		halflt1 11250	. . . . . . . 8 |- ( 1 / 2 ) < 1
12	10, 11	eqbrtri 4674	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
13	12	a1i 11	. . . . . 6 |- ( A e. CC -> ( abs ` ( 1 / 2 ) ) < 1 )
14	4, 13	expcnv 14596	. . . . 5 |- ( A e. CC -> ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ~~> 0 )
15		id 22	. . . . 5 |- ( A e. CC -> A e. CC )
16		geo2lim.1	. . . . . . 7 |- F = ( k e. NN |-> ( A / ( 2 ^ k ) ) )
17		nnex 11026	. . . . . . . 8 |- NN e. _V
18	17	mptex 6486	. . . . . . 7 |- ( k e. NN |-> ( A / ( 2 ^ k ) ) ) e. _V
19	16, 18	eqeltri 2697	. . . . . 6 |- F e. _V
20	19	a1i 11	. . . . 5 |- ( A e. CC -> F e. _V )
21		nnnn0 11299	. . . . . . . . 9 |- ( j e. NN -> j e. NN0 )
22	21	adantl 482	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> j e. NN0 )
23		oveq2 6658	. . . . . . . . 9 |- ( k = j -> ( ( 1 / 2 ) ^ k ) = ( ( 1 / 2 ) ^ j ) )
24		eqid 2622	. . . . . . . . 9 |- ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) = ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) )
25		ovex 6678	. . . . . . . . 9 |- ( ( 1 / 2 ) ^ j ) e. _V
26	23, 24, 25	fvmpt 6282	. . . . . . . 8 |- ( j e. NN0 -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
27	22, 26	syl 17	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
28		nnz 11399	. . . . . . . . 9 |- ( j e. NN -> j e. ZZ )
29	28	adantl 482	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> j e. ZZ )
30		2cn 11091	. . . . . . . . 9 |- 2 e. CC
31		2ne0 11113	. . . . . . . . 9 |- 2 =/= 0
32		exprec 12901	. . . . . . . . 9 |- ( ( 2 e. CC /\ 2 =/= 0 /\ j e. ZZ ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
33	30, 31, 32	mp3an12 1414	. . . . . . . 8 |- ( j e. ZZ -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
34	29, 33	syl 17	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
35	27, 34	eqtrd 2656	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
36		2nn 11185	. . . . . . . . 9 |- 2 e. NN
37		nnexpcl 12873	. . . . . . . . 9 |- ( ( 2 e. NN /\ j e. NN0 ) -> ( 2 ^ j ) e. NN )
38	36, 22, 37	sylancr 695	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. NN )
39	38	nnrecred 11066	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 1 / ( 2 ^ j ) ) e. RR )
40	39	recnd 10068	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( 1 / ( 2 ^ j ) ) e. CC )
41	35, 40	eqeltrd 2701	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) e. CC )
42		simpl 473	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> A e. CC )
43	38	nncnd 11036	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. CC )
44	38	nnne0d 11065	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) =/= 0 )
45	42, 43, 44	divrecd 10804	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) = ( A x. ( 1 / ( 2 ^ j ) ) ) )
46		oveq2 6658	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
47	46	oveq2d 6666	. . . . . . . 8 |- ( k = j -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ j ) ) )
48		ovex 6678	. . . . . . . 8 |- ( A / ( 2 ^ j ) ) e. _V
49	47, 16, 48	fvmpt 6282	. . . . . . 7 |- ( j e. NN -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
50	49	adantl 482	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
51	35	oveq2d 6666	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( A x. ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) ) = ( A x. ( 1 / ( 2 ^ j ) ) ) )
52	45, 50, 51	3eqtr4d 2666	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) = ( A x. ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) ) )
53	1, 2, 14, 15, 20, 41, 52	climmulc2 14367	. . . 4 |- ( A e. CC -> F ~~> ( A x. 0 ) )
54		mul01 10215	. . . 4 |- ( A e. CC -> ( A x. 0 ) = 0 )
55	53, 54	breqtrd 4679	. . 3 |- ( A e. CC -> F ~~> 0 )
56		seqex 12803	. . . 4 |- seq 1 ( + , F ) e. _V
57	56	a1i 11	. . 3 |- ( A e. CC -> seq 1 ( + , F ) e. _V )
58	42, 43, 44	divcld 10801	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) e. CC )
59	50, 58	eqeltrd 2701	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) e. CC )
60	50	oveq2d 6666	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A - ( F ` j ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
61		geo2sum 14604	. . . . 5 |- ( ( j e. NN /\ A e. CC ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
62	61	ancoms 469	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
63		elfznn 12370	. . . . . . 7 |- ( n e. ( 1 ... j ) -> n e. NN )
64	63	adantl 482	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> n e. NN )
65		oveq2 6658	. . . . . . . 8 |- ( k = n -> ( 2 ^ k ) = ( 2 ^ n ) )
66	65	oveq2d 6666	. . . . . . 7 |- ( k = n -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ n ) ) )
67		ovex 6678	. . . . . . 7 |- ( A / ( 2 ^ n ) ) e. _V
68	66, 16, 67	fvmpt 6282	. . . . . 6 |- ( n e. NN -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
69	64, 68	syl 17	. . . . 5 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
70		simpr 477	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> j e. NN )
71	70, 1	syl6eleq 2711	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> j e. ( ZZ>= ` 1 ) )
72		simpll 790	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> A e. CC )
73		nnnn0 11299	. . . . . . . . 9 |- ( n e. NN -> n e. NN0 )
74		nnexpcl 12873	. . . . . . . . 9 |- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN )
75	36, 73, 74	sylancr 695	. . . . . . . 8 |- ( n e. NN -> ( 2 ^ n ) e. NN )
76	64, 75	syl 17	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> ( 2 ^ n ) e. NN )
77	76	nncnd 11036	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> ( 2 ^ n ) e. CC )
78	76	nnne0d 11065	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> ( 2 ^ n ) =/= 0 )
79	72, 77, 78	divcld 10801	. . . . 5 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( 1 ... j ) ) -> ( A / ( 2 ^ n ) ) e. CC )
80	69, 71, 79	fsumser 14461	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( seq 1 ( + , F ) ` j ) )
81	60, 62, 80	3eqtr2rd 2663	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( seq 1 ( + , F ) ` j ) = ( A - ( F ` j ) ) )
82	1, 2, 55, 15, 57, 59, 81	climsubc2 14369	. 2 |- ( A e. CC -> seq 1 ( + , F ) ~~> ( A - 0 ) )
83		subid1 10301	. 2 |- ( A e. CC -> ( A - 0 ) = A )
84	82, 83	breqtrd 4679	1 |- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  omssubadd  30362  sge0ad2en  40648