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Mirrors > Home > MPE Home > Th. List > pntibndlem2a | Structured version Visualization version GIF version |
Description: Lemma for pntibndlem2 27649. (Contributed by Mario Carneiro, 7-Jun-2016.) |
Ref | Expression |
---|---|
pntibnd.r | ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) |
pntibndlem1.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
pntibndlem1.l | ⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) |
pntibndlem3.2 | ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴) |
pntibndlem3.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
pntibndlem3.k | ⊢ 𝐾 = (exp‘(𝐵 / (𝐸 / 2))) |
pntibndlem3.c | ⊢ 𝐶 = ((2 · 𝐵) + (log‘2)) |
pntibndlem3.4 | ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) |
pntibndlem3.6 | ⊢ (𝜑 → 𝑍 ∈ ℝ+) |
pntibndlem2.10 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
pntibndlem2a | ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁))) → (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pntibndlem2.10 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
2 | 1 | nnred 12278 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | 1red 11259 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
4 | ioossre 13444 | . . . . . . 7 ⊢ (0(,)1) ⊆ ℝ | |
5 | pntibnd.r | . . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) | |
6 | pntibndlem1.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
7 | pntibndlem1.l | . . . . . . . 8 ⊢ 𝐿 = ((1 / 4) / (𝐴 + 3)) | |
8 | 5, 6, 7 | pntibndlem1 27647 | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |
9 | 4, 8 | sselid 3992 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
10 | pntibndlem3.4 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) | |
11 | 4, 10 | sselid 3992 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
12 | 9, 11 | remulcld 11288 | . . . . 5 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ) |
13 | 3, 12 | readdcld 11287 | . . . 4 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ) |
14 | 13, 2 | remulcld 11288 | . . 3 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑁) ∈ ℝ) |
15 | elicc2 13448 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ ((1 + (𝐿 · 𝐸)) · 𝑁) ∈ ℝ) → (𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁)) ↔ (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁)))) | |
16 | 2, 14, 15 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁)) ↔ (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁)))) |
17 | 16 | biimpa 476 | 1 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁))) → (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ≤ cle 11293 − cmin 11489 / cdiv 11917 ℕcn 12263 2c2 12318 3c3 12319 4c4 12320 ℝ+crp 13031 (,)cioo 13383 [,]cicc 13386 abscabs 15269 expce 16093 logclog 26610 ψcchp 27150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-rp 13032 df-ioo 13387 df-icc 13390 |
This theorem is referenced by: pntibndlem2 27649 |
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