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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem13 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 33107. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem13.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem13.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem13.1 | ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) |
Ref | Expression |
---|---|
knoppndvlem13 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem13.1 | . . . 4 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) | |
2 | 1 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 0) → 1 < (𝑁 · (abs‘𝐶))) |
3 | 0lt1 10897 | . . . . . 6 ⊢ 0 < 1 | |
4 | 0re 10378 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | 1re 10376 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | ltnsymi 10495 | . . . . . 6 ⊢ (0 < 1 → ¬ 1 < 0) |
7 | 3, 6 | ax-mp 5 | . . . . 5 ⊢ ¬ 1 < 0 |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 0) → ¬ 1 < 0) |
9 | id 22 | . . . . . . . . . 10 ⊢ (𝐶 = 0 → 𝐶 = 0) | |
10 | 9 | abs00bd 14438 | . . . . . . . . 9 ⊢ (𝐶 = 0 → (abs‘𝐶) = 0) |
11 | 10 | oveq2d 6938 | . . . . . . . 8 ⊢ (𝐶 = 0 → (𝑁 · (abs‘𝐶)) = (𝑁 · 0)) |
12 | 11 | adantl 475 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · (abs‘𝐶)) = (𝑁 · 0)) |
13 | knoppndvlem13.n | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
14 | nncn 11383 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
15 | 13, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
16 | 15 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝑁 ∈ ℂ) |
17 | 16 | mul01d 10575 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · 0) = 0) |
18 | 12, 17 | eqtrd 2813 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 0) → (𝑁 · (abs‘𝐶)) = 0) |
19 | 18 | eqcomd 2783 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 0) → 0 = (𝑁 · (abs‘𝐶))) |
20 | 19 | breq2d 4898 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 0) → (1 < 0 ↔ 1 < (𝑁 · (abs‘𝐶)))) |
21 | 8, 20 | mtbid 316 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 0) → ¬ 1 < (𝑁 · (abs‘𝐶))) |
22 | 2, 21 | pm2.65da 807 | . 2 ⊢ (𝜑 → ¬ 𝐶 = 0) |
23 | 22 | neqned 2975 | 1 ⊢ (𝜑 → 𝐶 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 · cmul 10277 < clt 10411 -cneg 10607 ℕcn 11374 (,)cioo 12487 abscabs 14381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 |
This theorem is referenced by: knoppndvlem14 33098 knoppndvlem17 33101 |
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