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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem20 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33873. (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 33862 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
5 | 4 | simprd 498 | . . 3 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
6 | 2re 11712 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ) |
8 | 2 | nnred 11653 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
9 | 7, 8 | remulcld 10671 | . . . . . . 7 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
10 | 1 | knoppndvlem3 33853 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
11 | 10 | simpld 497 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
12 | 11 | recnd 10669 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
13 | 12 | abscld 14796 | . . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
14 | 9, 13 | remulcld 10671 | . . . . . 6 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
15 | 1red 10642 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
16 | 14, 15 | resubcld 11068 | . . . . 5 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ) |
17 | 0red 10644 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 0lt1 11162 | . . . . . . 7 ⊢ 0 < 1 | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 1) |
20 | 17, 15, 16, 19, 5 | lttrd 10801 | . . . . 5 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
21 | 16, 20 | elrpd 12429 | . . . 4 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ+) |
22 | 21 | recgt1d 12446 | . . 3 ⊢ (𝜑 → (1 < (((2 · 𝑁) · (abs‘𝐶)) − 1) ↔ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1)) |
23 | 5, 22 | mpbid 234 | . 2 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) < 1) |
24 | 21 | rprecred 12443 | . . . 4 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ) |
25 | 24, 15 | jca 514 | . . 3 ⊢ (𝜑 → ((1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ ∧ 1 ∈ ℝ)) |
26 | difrp 12428 | . . 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 234 | 1 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 < clt 10675 − cmin 10870 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 ℝ+crp 12390 (,)cioo 12739 abscabs 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ioo 12743 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 |
This theorem is referenced by: knoppndvlem21 33871 knoppndvlem22 33872 |
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