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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 1909 | . 2 ⊢ Ⅎ𝑛𝜑 | |
2 | 0xr 11293 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*) |
4 | pnfxr 11300 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*) |
6 | rge0ssre 13468 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
7 | sge0ad2en.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) | |
8 | 6, 7 | sselid 3974 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | 2re 12319 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ) |
12 | nnnn0 12512 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
13 | 12 | adantl 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
14 | 11, 13 | reexpcld 14163 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ) |
15 | 2cnd 12323 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ) | |
16 | 2ne0 12349 | . . . . . . 7 ⊢ 2 ≠ 0 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ≠ 0) |
18 | 13 | nn0zd 12617 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
19 | 15, 17, 18 | expne0d 14152 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0) |
20 | 9, 14, 19 | redivcld 12075 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ) |
21 | 20 | rexrd 11296 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ*) |
22 | 2rp 13014 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
23 | 22 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ+) |
24 | 23, 18 | rpexpcld 14245 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+) |
25 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
26 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | icogelb 13410 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,)+∞)) → 0 ≤ 𝐴) | |
28 | 25, 26, 7, 27 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴) |
29 | 28 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝐴) |
30 | 9, 24, 29 | divge0d 13091 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐴 / (2↑𝑛))) |
31 | 20 | ltpnfd 13136 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) < +∞) |
32 | 3, 5, 21, 30, 31 | elicod 13409 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ (0[,)+∞)) |
33 | 1zzd 12626 | . 2 ⊢ (𝜑 → 1 ∈ ℤ) | |
34 | nnuz 12898 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
35 | 8 | recnd 11274 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
36 | eqid 2725 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) | |
37 | 36 | geo2lim 15857 | . . 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 45953 | 1 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 +∞cpnf 11277 ℝ*cxr 11279 ≤ cle 11281 / cdiv 11903 ℕcn 12245 2c2 12300 ℕ0cn0 12505 ℝ+crp 13009 [,)cico 13361 seqcseq 14002 ↑cexp 14062 ⇝ cli 15464 Σ^csumge0 45885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-sumge0 45886 |
This theorem is referenced by: ovnsubaddlem1 46093 ovolval5lem1 46175 |
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