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| Mirrors > Home > MPE Home > Th. List > pntibndlem2a | Structured version Visualization version GIF version | ||
| Description: Lemma for pntibndlem2 27635. (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 12281 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 3 | 1red 11262 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 4 | ioossre 13448 | . . . . . . 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 27633 | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |
| 9 | 4, 8 | sselid 3981 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
| 10 | pntibndlem3.4 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) | |
| 11 | 4, 10 | sselid 3981 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 12 | 9, 11 | remulcld 11291 | . . . . 5 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ) |
| 13 | 3, 12 | readdcld 11290 | . . . 4 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ) |
| 14 | 13, 2 | remulcld 11291 | . . 3 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑁) ∈ ℝ) |
| 15 | elicc2 13452 | . . 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 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 4c4 12323 ℝ+crp 13034 (,)cioo 13387 [,]cicc 13390 abscabs 15273 expce 16097 logclog 26596 ψcchp 27136 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-rp 13035 df-ioo 13391 df-icc 13394 |
| This theorem is referenced by: pntibndlem2 27635 |
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