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Mirrors > Home > MPE Home > Th. List > pntibndlem2a | Structured version Visualization version GIF version |
Description: Lemma for pntibndlem2 25080. (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 10912 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | 1red 9934 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
4 | ioossre 12106 | . . . . . . 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 25078 | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |
9 | 4, 8 | sseldi 3566 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
10 | pntibndlem3.4 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) | |
11 | 4, 10 | sseldi 3566 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
12 | 9, 11 | remulcld 9949 | . . . . 5 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ) |
13 | 3, 12 | readdcld 9948 | . . . 4 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ) |
14 | 13, 2 | remulcld 9949 | . . 3 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑁) ∈ ℝ) |
15 | elicc2 12109 | . . 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 691 | . 2 ⊢ (𝜑 → (𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁)) ↔ (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁)))) |
17 | 16 | biimpa 500 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 4c4 10949 ℝ+crp 11708 (,)cioo 12046 [,]cicc 12049 abscabs 13822 expce 14631 logclog 24105 ψcchp 24619 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 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-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-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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 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-4 10958 df-rp 11709 df-ioo 12050 df-icc 12053 |
This theorem is referenced by: pntibndlem2 25080 |
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