Theorem geo2lim 11317

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 9385	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 9105	. . 3 |- ( A e. CC -> 1 e. ZZ )
3		halfcn 8958	. . . . . . 7 |- ( 1 / 2 ) e. CC
4	3	a1i 9	. . . . . 6 |- ( A e. CC -> ( 1 / 2 ) e. CC )
5		halfre 8957	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
6		halfge0 8960	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
7		absid 10875	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
8	5, 6, 7	mp2an 423	. . . . . . . 8 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
9		halflt1 8961	. . . . . . . 8 |- ( 1 / 2 ) < 1
10	8, 9	eqbrtri 3957	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
11	10	a1i 9	. . . . . 6 |- ( A e. CC -> ( abs ` ( 1 / 2 ) ) < 1 )
12	4, 11	expcnv 11305	. . . . 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 8750	. . . . . . . 8 |- NN e. _V
16	15	mptex 5654	. . . . . . 7 |- ( k e. NN |-> ( A / ( 2 ^ k ) ) ) e. _V
17	14, 16	eqeltri 2213	. . . . . 6 |- F e. _V
18	17	a1i 9	. . . . 5 |- ( A e. CC -> F e. _V )
19		nnnn0 9008	. . . . . . . 8 |- ( j e. NN -> j e. NN0 )
20	19	adantl 275	. . . . . . 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 10455	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) e. CC )
23		oveq2 5790	. . . . . . . 8 |- ( k = j -> ( ( 1 / 2 ) ^ k ) = ( ( 1 / 2 ) ^ j ) )
24		eqid 2140	. . . . . . . 8 |- ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) = ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) )
25	23, 24	fvmptg 5505	. . . . . . 7 |- ( ( j e. NN0 /\ ( ( 1 / 2 ) ^ j ) e. CC ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
26	20, 22, 25	syl2anc 409	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
27	26, 22	eqeltrd 2217	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) e. CC )
28		simpl 108	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> A e. CC )
29		2nn 8905	. . . . . . . . 9 |- 2 e. NN
30		nnexpcl 10337	. . . . . . . . 9 |- ( ( 2 e. NN /\ j e. NN0 ) -> ( 2 ^ j ) e. NN )
31	29, 20, 30	sylancr 411	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. NN )
32	31	nncnd 8758	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) e. CC )
33	31	nnap0d 8790	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( 2 ^ j ) # 0 )
34	28, 32, 33	divrecapd 8577	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) = ( A x. ( 1 / ( 2 ^ j ) ) ) )
35		simpr 109	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> j e. NN )
36	28, 32, 33	divclapd 8574	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( A / ( 2 ^ j ) ) e. CC )
37		oveq2 5790	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
38	37	oveq2d 5798	. . . . . . . 8 |- ( k = j -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ j ) ) )
39	38, 14	fvmptg 5505	. . . . . . 7 |- ( ( j e. NN /\ ( A / ( 2 ^ j ) ) e. CC ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
40	35, 36, 39	syl2anc 409	. . . . . 6 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) = ( A / ( 2 ^ j ) ) )
41		2cn 8815	. . . . . . . . 9 |- 2 e. CC
42		2ap0 8837	. . . . . . . . 9 |- 2 # 0
43		nnz 9097	. . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
44	43	adantl 275	. . . . . . . . 9 |- ( ( A e. CC /\ j e. NN ) -> j e. ZZ )
45		exprecap 10365	. . . . . . . . 9 |- ( ( 2 e. CC /\ 2 # 0 /\ j e. ZZ ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
46	41, 42, 44, 45	mp3an12i 1320	. . . . . . . 8 |- ( ( A e. CC /\ j e. NN ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
47	26, 46	eqtrd 2173	. . . . . . 7 |- ( ( A e. CC /\ j e. NN ) -> ( ( k e. NN0 |-> ( ( 1 / 2 ) ^ k ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
48	47	oveq2d 5798	. . . . . 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 2183	. . . . 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 11132	. . . 4 |- ( A e. CC -> F ~~> ( A x. 0 ) )
51		mul01 8175	. . . 4 |- ( A e. CC -> ( A x. 0 ) = 0 )
52	50, 51	breqtrd 3962	. . 3 |- ( A e. CC -> F ~~> 0 )
53		seqex 10251	. . . 4 |- seq 1 ( + , F ) e. _V
54	53	a1i 9	. . 3 |- ( A e. CC -> seq 1 ( + , F ) e. _V )
55	40, 36	eqeltrd 2217	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( F ` j ) e. CC )
56	40	oveq2d 5798	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> ( A - ( F ` j ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
57		geo2sum 11315	. . . . 5 |- ( ( j e. NN /\ A e. CC ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
58	57	ancoms 266	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( A - ( A / ( 2 ^ j ) ) ) )
59		elnnuz 9386	. . . . . . . 8 |- ( n e. NN <-> n e. ( ZZ>= ` 1 ) )
60	59	biimpri 132	. . . . . . 7 |- ( n e. ( ZZ>= ` 1 ) -> n e. NN )
61	60	adantl 275	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN )
62		simpll 519	. . . . . . 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 9054	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN0 )
65	63, 64	expcld 10455	. . . . . . 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 9196	. . . . . . . 8 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> n e. ZZ )
68	63, 66, 67	expap0d 10461	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( 2 ^ n ) # 0 )
69	62, 65, 68	divclapd 8574	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( A / ( 2 ^ n ) ) e. CC )
70		oveq2 5790	. . . . . . . 8 |- ( k = n -> ( 2 ^ k ) = ( 2 ^ n ) )
71	70	oveq2d 5798	. . . . . . 7 |- ( k = n -> ( A / ( 2 ^ k ) ) = ( A / ( 2 ^ n ) ) )
72	71, 14	fvmptg 5505	. . . . . 6 |- ( ( n e. NN /\ ( A / ( 2 ^ n ) ) e. CC ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
73	61, 69, 72	syl2anc 409	. . . . 5 |- ( ( ( A e. CC /\ j e. NN ) /\ n e. ( ZZ>= ` 1 ) ) -> ( F ` n ) = ( A / ( 2 ^ n ) ) )
74	35, 1	eleqtrdi 2233	. . . . 5 |- ( ( A e. CC /\ j e. NN ) -> j e. ( ZZ>= ` 1 ) )
75	73, 74, 69	fsum3ser 11198	. . . 4 |- ( ( A e. CC /\ j e. NN ) -> sum_ n e. ( 1 ... j ) ( A / ( 2 ^ n ) ) = ( seq 1 ( + , F ) ` j ) )
76	56, 58, 75	3eqtr2rd 2180	. . 3 |- ( ( A e. CC /\ j e. NN ) -> ( seq 1 ( + , F ) ` j ) = ( A - ( F ` j ) ) )
77	1, 2, 52, 13, 54, 55, 76	climsubc2 11134	. 2 |- ( A e. CC -> seq 1 ( + , F ) ~~> ( A - 0 ) )
78		subid1 8006	. 2 |- ( A e. CC -> ( A - 0 ) = A )
79	77, 78	breqtrd 3962	1 |- ( A e. CC -> seq 1 ( + , F ) ~~> A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   class class class wbr 3937   |-> cmpt 3997   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   < clt 7824   <_ cle 7825   - cmin 7957   # cap 8367   / cdiv 8456   NNcn 8744   2c2 8795   NN0cn0 9001   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821   seqcseq 10249   ^cexp 10323   abscabs 10801   ~~> cli 11079   sum_csu 11154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by:  trilpolemeq1  13408