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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem12 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33868. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem12.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem12.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem12.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndvlem12 | ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 10636 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
2 | 2re 11705 | . . . . . 6 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | knoppndvlem12.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | nnre 11639 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 3, 6 | remulcld 10665 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
8 | knoppndvlem12.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
9 | 8 | knoppndvlem3 33848 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 497 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 10 | recnd 10663 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 11 | abscld 14790 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
13 | 7, 12 | remulcld 10665 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
14 | 1lt2 11802 | . . . . . 6 ⊢ 1 < 2 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 < 2) |
16 | 2t1e2 11794 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
17 | 16 | eqcomi 2830 | . . . . . . . 8 ⊢ 2 = (2 · 1) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 = (2 · 1)) |
19 | 6, 12 | remulcld 10665 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 · (abs‘𝐶)) ∈ ℝ) |
20 | 2rp 12388 | . . . . . . . . 9 ⊢ 2 ∈ ℝ+ | |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ+) |
22 | knoppndvlem12.1 | . . . . . . . 8 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
23 | 1, 19, 21, 22 | ltmul2dd 12481 | . . . . . . 7 ⊢ (𝜑 → (2 · 1) < (2 · (𝑁 · (abs‘𝐶)))) |
24 | 18, 23 | eqbrtrd 5080 | . . . . . 6 ⊢ (𝜑 → 2 < (2 · (𝑁 · (abs‘𝐶)))) |
25 | 3 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ) |
26 | 6 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | 12 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ) |
28 | 25, 26, 27 | mulassd 10658 | . . . . . . 7 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) = (2 · (𝑁 · (abs‘𝐶)))) |
29 | 28 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑁 · (abs‘𝐶))) = ((2 · 𝑁) · (abs‘𝐶))) |
30 | 24, 29 | breqtrd 5084 | . . . . 5 ⊢ (𝜑 → 2 < ((2 · 𝑁) · (abs‘𝐶))) |
31 | 1, 3, 13, 15, 30 | lttrd 10795 | . . . 4 ⊢ (𝜑 → 1 < ((2 · 𝑁) · (abs‘𝐶))) |
32 | 1, 31 | jca 514 | . . 3 ⊢ (𝜑 → (1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶)))) |
33 | ltne 10731 | . . 3 ⊢ ((1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶))) → ((2 · 𝑁) · (abs‘𝐶)) ≠ 1) | |
34 | 32, 33 | syl 17 | . 2 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ≠ 1) |
35 | 1p1e2 11756 | . . . . 5 ⊢ (1 + 1) = 2 | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 + 1) = 2) |
37 | 36, 30 | eqbrtrd 5080 | . . 3 ⊢ (𝜑 → (1 + 1) < ((2 · 𝑁) · (abs‘𝐶))) |
38 | 1, 1, 13 | ltaddsubd 11234 | . . 3 ⊢ (𝜑 → ((1 + 1) < ((2 · 𝑁) · (abs‘𝐶)) ↔ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
39 | 37, 38 | mpbid 234 | . 2 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
40 | 34, 39 | jca 514 | 1 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 − cmin 10864 -cneg 10865 ℕcn 11632 2c2 11686 ℝ+crp 12383 (,)cioo 12732 abscabs 14587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ioo 12736 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 |
This theorem is referenced by: knoppndvlem14 33859 knoppndvlem15 33860 knoppndvlem17 33862 knoppndvlem20 33865 |
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