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 1922 | . 2 ⊢ Ⅎ𝑛𝜑 | |
2 | 0xr 10863 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*) |
4 | pnfxr 10870 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*) |
6 | rge0ssre 13027 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
7 | sge0ad2en.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) | |
8 | 6, 7 | sseldi 3889 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | 2re 11887 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ) |
12 | nnnn0 12080 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
13 | 12 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
14 | 11, 13 | reexpcld 13716 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ) |
15 | 2cnd 11891 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ) | |
16 | 2ne0 11917 | . . . . . . 7 ⊢ 2 ≠ 0 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ≠ 0) |
18 | 13 | nn0zd 12263 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
19 | 15, 17, 18 | expne0d 13705 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0) |
20 | 9, 14, 19 | redivcld 11643 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ) |
21 | 20 | rexrd 10866 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ*) |
22 | 2rp 12574 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
23 | 22 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ+) |
24 | 23, 18 | rpexpcld 13797 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+) |
25 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
26 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | icogelb 12969 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,)+∞)) → 0 ≤ 𝐴) | |
28 | 25, 26, 7, 27 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴) |
29 | 28 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝐴) |
30 | 9, 24, 29 | divge0d 12651 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐴 / (2↑𝑛))) |
31 | 20 | ltpnfd 12696 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) < +∞) |
32 | 3, 5, 21, 30, 31 | elicod 12968 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ (0[,)+∞)) |
33 | 1zzd 12191 | . 2 ⊢ (𝜑 → 1 ∈ ℤ) | |
34 | nnuz 12460 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
35 | 8 | recnd 10844 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
36 | eqid 2734 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) | |
37 | 36 | geo2lim 15420 | . . 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 43597 | 1 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 +∞cpnf 10847 ℝ*cxr 10849 ≤ cle 10851 / cdiv 11472 ℕcn 11813 2c2 11868 ℕ0cn0 12073 ℝ+crp 12569 [,)cico 12920 seqcseq 13557 ↑cexp 13618 ⇝ cli 15028 Σ^csumge0 43529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-rlim 15033 df-sum 15233 df-sumge0 43530 |
This theorem is referenced by: ovnsubaddlem1 43737 ovolval5lem1 43819 |
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