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Mirrors > Home > MPE Home > Th. List > pntibndlem2a | Structured version Visualization version GIF version |
Description: Lemma for pntibndlem2 25280. (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 11035 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
3 | 1red 10055 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
4 | ioossre 12235 | . . . . . . 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 25278 | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |
9 | 4, 8 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
10 | pntibndlem3.4 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) | |
11 | 4, 10 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
12 | 9, 11 | remulcld 10070 | . . . . 5 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ) |
13 | 3, 12 | readdcld 10069 | . . . 4 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ) |
14 | 13, 2 | remulcld 10070 | . . 3 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑁) ∈ ℝ) |
15 | elicc2 12238 | . . 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 693 | . 2 ⊢ (𝜑 → (𝑢 ∈ (𝑁[,]((1 + (𝐿 · 𝐸)) · 𝑁)) ↔ (𝑢 ∈ ℝ ∧ 𝑁 ≤ 𝑢 ∧ 𝑢 ≤ ((1 + (𝐿 · 𝐸)) · 𝑁)))) |
17 | 16 | biimpa 501 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 ℝ+crp 11832 (,)cioo 12175 [,]cicc 12178 abscabs 13974 expce 14792 logclog 24301 ψcchp 24819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-rp 11833 df-ioo 12179 df-icc 12182 |
This theorem is referenced by: pntibndlem2 25280 |
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