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Mirrors > Home > MPE Home > Th. List > recnz | Structured version Visualization version GIF version |
Description: The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.) |
Ref | Expression |
---|---|
recnz | ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ¬ (1 / 𝐴) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recgt1i 11274 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (0 < (1 / 𝐴) ∧ (1 / 𝐴) < 1)) | |
2 | 1 | simprd 491 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (1 / 𝐴) < 1) |
3 | 1 | simpld 490 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 0 < (1 / 𝐴)) |
4 | zgt0ge1 11783 | . . . 4 ⊢ ((1 / 𝐴) ∈ ℤ → (0 < (1 / 𝐴) ↔ 1 ≤ (1 / 𝐴))) | |
5 | 3, 4 | syl5ibcom 237 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ((1 / 𝐴) ∈ ℤ → 1 ≤ (1 / 𝐴))) |
6 | 1re 10376 | . . . 4 ⊢ 1 ∈ ℝ | |
7 | 0lt1 10897 | . . . . . . . 8 ⊢ 0 < 1 | |
8 | 0re 10378 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
9 | lttr 10453 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝐴) → 0 < 𝐴)) | |
10 | 8, 6, 9 | mp3an12 1524 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 < 𝐴) → 0 < 𝐴)) |
11 | 7, 10 | mpani 686 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (1 < 𝐴 → 0 < 𝐴)) |
12 | 11 | imdistani 564 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
13 | gt0ne0 10840 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 𝐴 ≠ 0) |
15 | rereccl 11093 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ) | |
16 | 14, 15 | syldan 585 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (1 / 𝐴) ∈ ℝ) |
17 | lenlt 10455 | . . . 4 ⊢ ((1 ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (1 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 1)) | |
18 | 6, 16, 17 | sylancr 581 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (1 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 1)) |
19 | 5, 18 | sylibd 231 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ((1 / 𝐴) ∈ ℤ → ¬ (1 / 𝐴) < 1)) |
20 | 2, 19 | mt2d 134 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ¬ (1 / 𝐴) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 < clt 10411 ≤ cle 10412 / cdiv 11032 ℤcz 11728 |
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 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 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 |
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 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 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-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 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-n0 11643 df-z 11729 |
This theorem is referenced by: halfnz 11807 facndiv 13393 dvdsprmpweqle 15994 |
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