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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem12 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 36516. (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 11259 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
2 | 2re 12337 | . . . . . 6 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ) |
4 | knoppndvlem12.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | nnre 12270 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
7 | 3, 6 | remulcld 11288 | . . . . . 6 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ) |
8 | knoppndvlem12.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
9 | 8 | knoppndvlem3 36496 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
10 | 9 | simpld 494 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
11 | 10 | recnd 11286 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 11 | abscld 15471 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ) |
13 | 7, 12 | remulcld 11288 | . . . . 5 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ) |
14 | 1lt2 12434 | . . . . . 6 ⊢ 1 < 2 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 < 2) |
16 | 2t1e2 12426 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
17 | 16 | eqcomi 2743 | . . . . . . . 8 ⊢ 2 = (2 · 1) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 = (2 · 1)) |
19 | 6, 12 | remulcld 11288 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 · (abs‘𝐶)) ∈ ℝ) |
20 | 2rp 13036 | . . . . . . . . 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 13130 | . . . . . . 7 ⊢ (𝜑 → (2 · 1) < (2 · (𝑁 · (abs‘𝐶)))) |
24 | 18, 23 | eqbrtrd 5169 | . . . . . 6 ⊢ (𝜑 → 2 < (2 · (𝑁 · (abs‘𝐶)))) |
25 | 3 | recnd 11286 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ) |
26 | 6 | recnd 11286 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | 12 | recnd 11286 | . . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ) |
28 | 25, 26, 27 | mulassd 11281 | . . . . . . 7 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) = (2 · (𝑁 · (abs‘𝐶)))) |
29 | 28 | eqcomd 2740 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑁 · (abs‘𝐶))) = ((2 · 𝑁) · (abs‘𝐶))) |
30 | 24, 29 | breqtrd 5173 | . . . . 5 ⊢ (𝜑 → 2 < ((2 · 𝑁) · (abs‘𝐶))) |
31 | 1, 3, 13, 15, 30 | lttrd 11419 | . . . 4 ⊢ (𝜑 → 1 < ((2 · 𝑁) · (abs‘𝐶))) |
32 | 1, 31 | jca 511 | . . 3 ⊢ (𝜑 → (1 ∈ ℝ ∧ 1 < ((2 · 𝑁) · (abs‘𝐶)))) |
33 | ltne 11355 | . . 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 12388 | . . . . 5 ⊢ (1 + 1) = 2 | |
36 | 35 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 + 1) = 2) |
37 | 36, 30 | eqbrtrd 5169 | . . 3 ⊢ (𝜑 → (1 + 1) < ((2 · 𝑁) · (abs‘𝐶))) |
38 | 1, 1, 13 | ltaddsubd 11860 | . . 3 ⊢ (𝜑 → ((1 + 1) < ((2 · 𝑁) · (abs‘𝐶)) ↔ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
39 | 37, 38 | mpbid 232 | . 2 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) |
40 | 34, 39 | jca 511 | 1 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 -cneg 11490 ℕcn 12263 2c2 12318 ℝ+crp 13031 (,)cioo 13383 abscabs 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ioo 13387 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 |
This theorem is referenced by: knoppndvlem14 36507 knoppndvlem15 36508 knoppndvlem17 36510 knoppndvlem20 36513 |
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