Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0ad2en | Structured version Visualization version GIF version |
Description: The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
sge0ad2en.1 | ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
sge0ad2en | ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . 2 ⊢ Ⅎ𝑛𝜑 | |
2 | 0xr 10687 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*) |
4 | pnfxr 10694 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*) |
6 | rge0ssre 12843 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
7 | sge0ad2en.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) | |
8 | 6, 7 | sseldi 3964 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | 2re 11710 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ) |
12 | nnnn0 11903 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
13 | 12 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
14 | 11, 13 | reexpcld 13526 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ) |
15 | 2cnd 11714 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ) | |
16 | 2ne0 11740 | . . . . . . 7 ⊢ 2 ≠ 0 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ≠ 0) |
18 | 13 | nn0zd 12084 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
19 | 15, 17, 18 | expne0d 13515 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0) |
20 | 9, 14, 19 | redivcld 11467 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ) |
21 | 20 | rexrd 10690 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ*) |
22 | 2rp 12393 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
23 | 22 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ+) |
24 | 23, 18 | rpexpcld 13607 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+) |
25 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
26 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | icogelb 12787 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,)+∞)) → 0 ≤ 𝐴) | |
28 | 25, 26, 7, 27 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴) |
29 | 28 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝐴) |
30 | 9, 24, 29 | divge0d 12470 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐴 / (2↑𝑛))) |
31 | 20 | ltpnfd 12515 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) < +∞) |
32 | 3, 5, 21, 30, 31 | elicod 12786 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ (0[,)+∞)) |
33 | 1zzd 12012 | . 2 ⊢ (𝜑 → 1 ∈ ℤ) | |
34 | nnuz 12280 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
35 | 8 | recnd 10668 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
36 | eqid 2821 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) | |
37 | 36 | geo2lim 15230 | . . 3 ⊢ (𝐴 ∈ ℂ → seq1( + , (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) ⇝ 𝐴) |
38 | 35, 37 | syl 17 | . 2 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) ⇝ 𝐴) |
39 | 1, 32, 33, 34, 38 | sge0isummpt 42711 | 1 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 +∞cpnf 10671 ℝ*cxr 10673 ≤ cle 10675 / cdiv 11296 ℕcn 11637 2c2 11691 ℕ0cn0 11896 ℝ+crp 12388 [,)cico 12739 seqcseq 13368 ↑cexp 13428 ⇝ cli 14840 Σ^csumge0 42643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 df-sum 15042 df-sumge0 42644 |
This theorem is referenced by: ovnsubaddlem1 42851 ovolval5lem1 42933 |
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