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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem20 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33052. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
Ref | Expression |
---|---|
knoppndvlem20.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem20.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem20.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndvlem20 | ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem20.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
2 | knoppndvlem20.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | knoppndvlem20.1 | . . . . 5 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
4 | 1, 2, 3 | knoppndvlem12 33041 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
5 | 4 | simprd 491 | . . 3 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
6 | 2re 11432 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ) |
8 | 2 | nnred 11374 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
9 | 7, 8 | remulcld 10394 | . . . . . . 7 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
10 | 1 | knoppndvlem3 33032 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
11 | 10 | simpld 490 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
12 | 11 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
13 | 12 | abscld 14559 | . . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
14 | 9, 13 | remulcld 10394 | . . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
15 | 1red 10364 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
16 | 14, 15 | resubcld 10789 | . . . . 5 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ) |
17 | 0red 10367 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 0lt1 10881 | . . . . . . 7 ⊢ 0 < 1 | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 1) |
20 | 17, 15, 16, 19, 5 | lttrd 10524 | . . . . 5 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
21 | 16, 20 | elrpd 12160 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ+) |
22 | 21 | recgt1d 12177 | . . 3 ⊢ (𝜑 → (1 < (((2 · 𝑁) · (abs‘𝐶)) − 1) ↔ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1)) |
23 | 5, 22 | mpbid 224 | . 2 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1) |
24 | 21 | rprecred 12174 | . . . 4 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ) |
25 | 24, 15 | jca 507 | . . 3 ⊢ (𝜑 → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ ∧ 1 ∈ ℝ)) |
26 | difrp 12159 | . . 3 ⊢ (((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1 ↔ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)) | |
27 | 25, 26 | syl 17 | . 2 ⊢ (𝜑 → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1 ↔ (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)) |
28 | 23, 27 | mpbid 224 | 1 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 0cc0 10259 1c1 10260 · cmul 10264 < clt 10398 − cmin 10592 -cneg 10593 / cdiv 11016 ℕcn 11357 2c2 11413 ℝ+crp 12119 (,)cioo 12470 abscabs 14358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-ioo 12474 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 |
This theorem is referenced by: knoppndvlem21 33050 knoppndvlem22 33051 |
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