Theorem geo2lim 12013

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 9746	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9461	. . 3 |- ( A e. CC -> 1 e. ZZ )
3		halfcn 9313	. . . . . . 7 |- ( 1 / 2 ) e. CC
4	3	a1i 9	. . . . . 6 |- ( A e. CC -> ( 1 / 2 ) e. CC )
5		halfre 9312	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
6		halfge0 9315	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
7		absid 11568	. . . . . . . . 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 9316	. . . . . . . 8 |- ( 1 / 2 ) < 1
10	8, 9	eqbrtri 4103	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
11	10	a1i 9	. . . . . 6 |- ( A e. CC -> ( abs ` ( 1 / 2 ) ) < 1 )
12	4, 11	expcnv 12001	. . . . 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 9104	. . . . . . . 8 |- NN e. _V
16	15	mptex 5858	. . . . . . 7 |- ( k e. NN |-> ( A / ( 2 ^ k ) ) ) e. _V
17	14, 16	eqeltri 2302	. . . . . 6 |- F e. _V
18	17	a1i 9	. . . . 5 |- ( A e. CC -> F e. _V )
19		nnnn0 9364	. . . . . . . 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 10882	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) e. CC )
23		oveq2 6002	. . . . . . . 8 |- ( k = j -> ( ( 1 / 2 ) ^ k ) = ( ( 1 / 2 ) ^ j ) )
24		eqid 2229	. . . . . . . 8 |- ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) = ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) )
25	23, 24	fvmptg 5703	. . . . . . 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 2306	. . . . 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 9260	. . . . . . . . 9 |- 2 e. NN
30		nnexpcl 10761	. . . . . . . . 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 9112	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. CC )
33	31	nnap0d 9144	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) # 0 )
34	28, 32, 33	divrecapd 8928	. . . . . 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 8925	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) e. CC )
37		oveq2 6002	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
38	37	oveq2d 6010	. . . . . . . 8 |- ( k = j -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ j ) ) )
39	38, 14	fvmptg 5703	. . . . . . 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 9169	. . . . . . . . 9 |- 2 e. CC
42		2ap0 9191	. . . . . . . . 9 |- 2 # 0
43		nnz 9453	. . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
44	43	adantl 277	. . . . . . . . 9 |- ( ( A e. CC /\ j e. NN ) -> j e. ZZ )
45		exprecap 10789	. . . . . . . . 9 |- ( ( 2 e. CC /\ 2 # 0 /\ j e. ZZ ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
46	41, 42, 44, 45	mp3an12i 1375	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
47	26, 46	eqtrd 2262	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
48	47	oveq2d 6010	. . . . . 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 2272	. . . . 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 11828	. . . 4 |- ( A e. CC -> F ~~> ( A x. 0 ) )
51		mul01 8523	. . . 4 |- ( A e. CC -> ( A x. 0 ) = 0 )
52	50, 51	breqtrd 4108	. . 3 |- ( A e. CC -> F ~~> 0 )
53		seqex 10658	. . . 4 |- seq 1 ( + , F ) e. _V
54	53	a1i 9	. . 3 |- ( A e. CC -> seq 1 ( + , F ) e. _V )
55	40, 36	eqeltrd 2306	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) e. CC )
56	40	oveq2d 6010	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A - ( F ` j ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
57		geo2sum 12011	. . . . 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 9747	. . . . . . . 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 9410	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN0 )
65	63, 64	expcld 10882	. . . . . . 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 9556	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. ZZ )
68	63, 66, 67	expap0d 10888	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( 2 ^ n ) # 0 )
69	62, 65, 68	divclapd 8925	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( A / ( 2 ^ n ) ) e. CC )
70		oveq2 6002	. . . . . . . 8 |- ( k = n -> ( 2 ^ k ) = ( 2 ^ n ) )
71	70	oveq2d 6010	. . . . . . 7 |- ( k = n -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ n ) ) )
72	71, 14	fvmptg 5703	. . . . . 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 2322	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> j e. ( ZZ>= ` 1 ) )
75	73, 74, 69	fsum3ser 11894	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( seq 1 ( + , F ) ` j ) )
76	56, 58, 75	3eqtr2rd 2269	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( seq 1 ( + , F ) ` j ) = ( A - ( F ` j ) ) )
77	1, 2, 52, 13, 54, 55, 76	climsubc2 11830	. 2 |- ( A e. CC -> seq 1 ( + , F ) ~~> ( A - 0 ) )
78		subid1 8354	. 2 |- ( A e. CC -> ( A - 0 ) = A )
79	77, 78	breqtrd 4108	1 |- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   class class class wbr 4082   |-> cmpt 4144   ` cfv 5314  (class class class)co 5994   CCcc 7985   RRcr 7986   0cc0 7987   1c1 7988   + caddc 7990   x. cmul 7992   < clt 8169   <_ cle 8170   - cmin 8305   # cap 8716   / cdiv 8807   NNcn 9098   2c2 9149   NN0cn0 9357   ZZcz 9434   ZZ>=cuz 9710   ...cfz 10192   seqcseq 10656   ^cexp 10747   abscabs 11494   ~~> cli 11775   sum_csu 11850

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-frec 6527  df-1o 6552  df-oadd 6556  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-clim 11776  df-sumdc 11851

This theorem is referenced by:  trilpolemeq1  16339