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 1830 | . 2 ⊢ Ⅎ𝑛𝜑 | |
2 | 0xr 9965 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*) |
4 | pnfxr 9971 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*) |
6 | rge0ssre 12151 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
7 | sge0ad2en.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) | |
8 | 6, 7 | sseldi 3566 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | 2re 10967 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ) |
12 | nnnn0 11176 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
13 | 12 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
14 | 11, 13 | reexpcld 12887 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ) |
15 | 2cnd 10970 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ) | |
16 | 2ne0 10990 | . . . . . . 7 ⊢ 2 ≠ 0 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ≠ 0) |
18 | 13 | nn0zd 11356 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
19 | 15, 17, 18 | expne0d 12876 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0) |
20 | 9, 14, 19 | redivcld 10732 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ) |
21 | 20 | rexrd 9968 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ*) |
22 | 2rp 11713 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
23 | 22 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ+) |
24 | 23, 18 | rpexpcld 12894 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+) |
25 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
26 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | icogelb 12096 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,)+∞)) → 0 ≤ 𝐴) | |
28 | 25, 26, 7, 27 | syl3anc 1318 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴) |
29 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝐴) |
30 | 9, 24, 29 | divge0d 11788 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐴 / (2↑𝑛))) |
31 | 20 | ltpnfd 11831 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) < +∞) |
32 | 3, 5, 21, 30, 31 | elicod 12095 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ (0[,)+∞)) |
33 | 1zzd 11285 | . 2 ⊢ (𝜑 → 1 ∈ ℤ) | |
34 | nnuz 11599 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
35 | 8 | recnd 9947 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
36 | eqid 2610 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) | |
37 | 36 | geo2lim 14445 | . . 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 39323 | 1 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℝ+crp 11708 [,)cico 12048 seqcseq 12663 ↑cexp 12722 ⇝ cli 14063 Σ^csumge0 39255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-sumge0 39256 |
This theorem is referenced by: ovnsubaddlem1 39460 ovolval5lem1 39542 |
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