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Mirrors > Home > MPE Home > Th. List > pntrlog2bndlem6a | Structured version Visualization version GIF version |
Description: Lemma for pntrlog2bndlem6 26495. (Contributed by Mario Carneiro, 7-Jun-2016.) |
Ref | Expression |
---|---|
pntsval.1 | ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) |
pntrlog2bnd.r | ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) |
pntrlog2bnd.t | ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0)) |
pntrlog2bndlem5.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
pntrlog2bndlem5.2 | ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵) |
pntrlog2bndlem6.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
pntrlog2bndlem6.2 | ⊢ (𝜑 → 1 ≤ 𝐴) |
Ref | Expression |
---|---|
pntrlog2bndlem6a | ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 12994 | . . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ) | |
2 | 1 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ) |
3 | 1rp 12619 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+) |
5 | 4 | rpred 12657 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ) |
6 | eliooord 13023 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞)) | |
7 | 6 | adantl 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞)) |
8 | 7 | simpld 498 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥) |
9 | 5, 2, 8 | ltled 11009 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥) |
10 | 2, 4, 9 | rpgecld 12696 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+) |
11 | pntrlog2bndlem6.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
12 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+) |
13 | pntrlog2bndlem6.2 | . . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝐴) | |
14 | 11, 12, 13 | rpgecld 12696 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
15 | 14 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ+) |
16 | 10, 15 | rpdivcld 12674 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ+) |
17 | 16 | rprege0d 12664 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑥 / 𝐴) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝐴))) |
18 | flge0nn0 13424 | . . . 4 ⊢ (((𝑥 / 𝐴) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝐴)) → (⌊‘(𝑥 / 𝐴)) ∈ ℕ0) | |
19 | nn0p1nn 12158 | . . . 4 ⊢ ((⌊‘(𝑥 / 𝐴)) ∈ ℕ0 → ((⌊‘(𝑥 / 𝐴)) + 1) ∈ ℕ) | |
20 | 17, 18, 19 | 3syl 18 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘(𝑥 / 𝐴)) + 1) ∈ ℕ) |
21 | nnuz 12506 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
22 | 20, 21 | eleqtrdi 2850 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘(𝑥 / 𝐴)) + 1) ∈ (ℤ≥‘1)) |
23 | 16 | rpred 12657 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ) |
24 | 10 | rpge0d 12661 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥) |
25 | 13 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝐴) |
26 | 4, 15, 2, 24, 25 | lediv2ad 12679 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ≤ (𝑥 / 1)) |
27 | 2 | recnd 10890 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ) |
28 | 27 | div1d 11629 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 1) = 𝑥) |
29 | 26, 28 | breqtrd 5095 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ≤ 𝑥) |
30 | flword2 13417 | . . 3 ⊢ (((𝑥 / 𝐴) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 / 𝐴) ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘(𝑥 / 𝐴)))) | |
31 | 23, 2, 29, 30 | syl3anc 1373 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘(𝑥 / 𝐴)))) |
32 | fzsplit2 13166 | . 2 ⊢ ((((⌊‘(𝑥 / 𝐴)) + 1) ∈ (ℤ≥‘1) ∧ (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘(𝑥 / 𝐴)))) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))) | |
33 | 22, 31, 32 | syl2anc 587 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3064 ∪ cun 3881 ifcif 4455 class class class wbr 5069 ↦ cmpt 5151 ‘cfv 6400 (class class class)co 7234 ℝcr 10757 0cc0 10758 1c1 10759 + caddc 10761 · cmul 10763 +∞cpnf 10893 < clt 10896 ≤ cle 10897 − cmin 11091 / cdiv 11518 ℕcn 11859 ℕ0cn0 12119 ℤ≥cuz 12467 ℝ+crp 12615 (,)cioo 12964 ...cfz 13124 ⌊cfl 13394 abscabs 14829 Σcsu 15281 logclog 25474 Λcvma 26005 ψcchp 26006 |
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 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 ax-pre-sup 10836 |
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 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-sup 9087 df-inf 9088 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-div 11519 df-nn 11860 df-n0 12120 df-z 12206 df-uz 12468 df-rp 12616 df-ioo 12968 df-fz 13125 df-fl 13396 |
This theorem is referenced by: pntrlog2bndlem6 26495 |
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