Theorem geo2lim 12206

Description: The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)

Hypothesis

Ref	Expression
geo2lim.1	⊢ 𝐹 = (𝑘 ∈ ℕ ↦ (𝐴 / (2↑𝑘)))

Assertion

Ref	Expression
geo2lim	⊢ (𝐴 ∈ ℂ → seq1( + , 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 9893	. . 3 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 9606	. . 3 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
3		halfcn 9454	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
4	3	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 / 2) ∈ ℂ)
5		halfre 9453	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
6		halfge0 9456	. . . . . . . . 9 ⊢ 0 ≤ (1 / 2)
7		absid 11760	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2)) → (abs‘(1 / 2)) = (1 / 2))
8	5, 6, 7	mp2an 426	. . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2)
9		halflt1 9457	. . . . . . . 8 ⊢ (1 / 2) < 1
10	8, 9	eqbrtri 4132	. . . . . . 7 ⊢ (abs‘(1 / 2)) < 1
11	10	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘(1 / 2)) < 1)
12	4, 11	expcnv 12194	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘)) ⇝ 0)
13		id 19	. . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
14		geo2lim.1	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ ↦ (𝐴 / (2↑𝑘)))
15		nnex 9245	. . . . . . . 8 ⊢ ℕ ∈ V
16	15	mptex 5914	. . . . . . 7 ⊢ (𝑘 ∈ ℕ ↦ (𝐴 / (2↑𝑘))) ∈ V
17	14, 16	eqeltri 2307	. . . . . 6 ⊢ 𝐹 ∈ V
18	17	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ V)
19		nnnn0 9505	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
20	19	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
21	3	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (1 / 2) ∈ ℂ)
22	21, 20	expcld 11039	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) ∈ ℂ)
23		oveq2 6060	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑗))
24		eqid 2234	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘)) = (𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))
25	23, 24	fvmptg 5755	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ0 ∧ ((1 / 2)↑𝑗) ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))‘𝑗) = ((1 / 2)↑𝑗))
26	20, 22, 25	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))‘𝑗) = ((1 / 2)↑𝑗))
27	26, 22	eqeltrd 2311	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))‘𝑗) ∈ ℂ)
28		simpl 109	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ ℂ)
29		2nn 9401	. . . . . . . . 9 ⊢ 2 ∈ ℕ
30		nnexpcl 10918	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
31	29, 20, 30	sylancr 414	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
32	31	nncnd 9253	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
33	31	nnap0d 9285	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (2↑𝑗) # 0)
34	28, 32, 33	divrecapd 9069	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐴 / (2↑𝑗)) = (𝐴 · (1 / (2↑𝑗))))
35		simpr 110	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
36	28, 32, 33	divclapd 9066	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐴 / (2↑𝑗)) ∈ ℂ)
37		oveq2 6060	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (2↑𝑘) = (2↑𝑗))
38	37	oveq2d 6068	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐴 / (2↑𝑘)) = (𝐴 / (2↑𝑗)))
39	38, 14	fvmptg 5755	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ ∧ (𝐴 / (2↑𝑗)) ∈ ℂ) → (𝐹‘𝑗) = (𝐴 / (2↑𝑗)))
40	35, 36, 39	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (𝐴 / (2↑𝑗)))
41		2cn 9310	. . . . . . . . 9 ⊢ 2 ∈ ℂ
42		2ap0 9332	. . . . . . . . 9 ⊢ 2 # 0
43		nnz 9598	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
44	43	adantl 277	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
45		exprecap 10946	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 # 0 ∧ 𝑗 ∈ ℤ) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
46	41, 42, 44, 45	mp3an12i 1378	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → ((1 / 2)↑𝑗) = (1 / (2↑𝑗)))
47	26, 46	eqtrd 2267	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))‘𝑗) = (1 / (2↑𝑗)))
48	47	oveq2d 6068	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐴 · ((𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))‘𝑗)) = (𝐴 · (1 / (2↑𝑗))))
49	34, 40, 48	3eqtr4d 2277	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (𝐴 · ((𝑘 ∈ ℕ0 ↦ ((1 / 2)↑𝑘))‘𝑗)))
50	1, 2, 12, 13, 18, 27, 49	climmulc2 12020	. . . 4 ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ (𝐴 · 0))
51		mul01 8664	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
52	50, 51	breqtrd 4137	. . 3 ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ 0)
53		seqex 10815	. . . 4 ⊢ seq1( + , 𝐹) ∈ V
54	53	a1i 9	. . 3 ⊢ (𝐴 ∈ ℂ → seq1( + , 𝐹) ∈ V)
55	40, 36	eqeltrd 2311	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ ℂ)
56	40	oveq2d 6068	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (𝐴 − (𝐹‘𝑗)) = (𝐴 − (𝐴 / (2↑𝑗))))
57		geo2sum 12204	. . . . 5 ⊢ ((𝑗 ∈ ℕ ∧ 𝐴 ∈ ℂ) → Σ𝑛 ∈ (1...𝑗)(𝐴 / (2↑𝑛)) = (𝐴 − (𝐴 / (2↑𝑗))))
58	57	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → Σ𝑛 ∈ (1...𝑗)(𝐴 / (2↑𝑛)) = (𝐴 − (𝐴 / (2↑𝑗))))
59		elnnuz 9894	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
60	59	biimpri 133	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘1) → 𝑛 ∈ ℕ)
61	60	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → 𝑛 ∈ ℕ)
62		simpll 527	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → 𝐴 ∈ ℂ)
63	41	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → 2 ∈ ℂ)
64	61	nnnn0d 9555	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → 𝑛 ∈ ℕ0)
65	63, 64	expcld 11039	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → (2↑𝑛) ∈ ℂ)
66	42	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → 2 # 0)
67	61	nnzd 9702	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → 𝑛 ∈ ℤ)
68	63, 66, 67	expap0d 11045	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → (2↑𝑛) # 0)
69	62, 65, 68	divclapd 9066	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → (𝐴 / (2↑𝑛)) ∈ ℂ)
70		oveq2 6060	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (2↑𝑘) = (2↑𝑛))
71	70	oveq2d 6068	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐴 / (2↑𝑘)) = (𝐴 / (2↑𝑛)))
72	71, 14	fvmptg 5755	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 / (2↑𝑛)) ∈ ℂ) → (𝐹‘𝑛) = (𝐴 / (2↑𝑛)))
73	61, 69, 72	syl2anc 411	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ≥‘1)) → (𝐹‘𝑛) = (𝐴 / (2↑𝑛)))
74	35, 1	eleqtrdi 2327	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ≥‘1))
75	73, 74, 69	fsum3ser 12087	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → Σ𝑛 ∈ (1...𝑗)(𝐴 / (2↑𝑛)) = (seq1( + , 𝐹)‘𝑗))
76	56, 58, 75	3eqtr2rd 2274	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) = (𝐴 − (𝐹‘𝑗)))
77	1, 2, 52, 13, 54, 55, 76	climsubc2 12022	. 2 ⊢ (𝐴 ∈ ℂ → seq1( + , 𝐹) ⇝ (𝐴 − 0))
78		subid1 8495	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
79	77, 78	breqtrd 4137	1 ⊢ (𝐴 ∈ ℂ → seq1( + , 𝐹) ⇝ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   class class class wbr 4111   ↦ cmpt 4173  ‘cfv 5354  (class class class)co 6052  ℂcc 8127  ℝcr 8128  0cc0 8129  1c1 8130   + caddc 8132   · cmul 8134   < clt 8310   ≤ cle 8311   − cmin 8446   # cap 8857   / cdiv 8948  ℕcn 9239  2c2 9290  ℕ₀cn0 9498  ℤcz 9579  ℤ_≥cuz 9856  ...cfz 10345  seqcseq 10813  ↑cexp 10904  abscabs 11686   ⇝ cli 11967  Σcsu 12042

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043

This theorem is referenced by:  trilpolemeq1  16841