Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rpgtrecnn | Structured version Visualization version GIF version |
Description: Any positive real number is greater than the reciprocal of a positive integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
rpgtrecnn | ⊢ (𝐴 ∈ ℝ+ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpreccl 12753 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) | |
2 | 1 | rpred 12769 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ) |
3 | 1 | rpge0d 12773 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (1 / 𝐴)) |
4 | flge0nn0 13536 | . . . 4 ⊢ (((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)) → (⌊‘(1 / 𝐴)) ∈ ℕ0) | |
5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (⌊‘(1 / 𝐴)) ∈ ℕ0) |
6 | nn0p1nn 12270 | . . 3 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℕ0 → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ) |
8 | flltp1 13516 | . . . . 5 ⊢ ((1 / 𝐴) ∈ ℝ → (1 / 𝐴) < ((⌊‘(1 / 𝐴)) + 1)) | |
9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) < ((⌊‘(1 / 𝐴)) + 1)) |
10 | 7 | nnrpd 12767 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ+) |
11 | 1, 10 | ltrecd 12787 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((1 / 𝐴) < ((⌊‘(1 / 𝐴)) + 1) ↔ (1 / ((⌊‘(1 / 𝐴)) + 1)) < (1 / (1 / 𝐴)))) |
12 | 9, 11 | mpbid 231 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (1 / ((⌊‘(1 / 𝐴)) + 1)) < (1 / (1 / 𝐴))) |
13 | rpcn 12737 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
14 | rpne0 12743 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
15 | 13, 14 | recrecd 11746 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (1 / (1 / 𝐴)) = 𝐴) |
16 | 12, 15 | breqtrd 5105 | . 2 ⊢ (𝐴 ∈ ℝ+ → (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴) |
17 | oveq2 7277 | . . . 4 ⊢ (𝑛 = ((⌊‘(1 / 𝐴)) + 1) → (1 / 𝑛) = (1 / ((⌊‘(1 / 𝐴)) + 1))) | |
18 | 17 | breq1d 5089 | . . 3 ⊢ (𝑛 = ((⌊‘(1 / 𝐴)) + 1) → ((1 / 𝑛) < 𝐴 ↔ (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴)) |
19 | 18 | rspcev 3561 | . 2 ⊢ ((((⌊‘(1 / 𝐴)) + 1) ∈ ℕ ∧ (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴) |
20 | 7, 16, 19 | syl2anc 584 | 1 ⊢ (𝐴 ∈ ℝ+ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 class class class wbr 5079 ‘cfv 6431 (class class class)co 7269 ℝcr 10869 0cc0 10870 1c1 10871 + caddc 10873 < clt 11008 ≤ cle 11009 / cdiv 11630 ℕcn 11971 ℕ0cn0 12231 ℝ+crp 12727 ⌊cfl 13506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-sup 9177 df-inf 9178 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12580 df-rp 12728 df-fl 13508 |
This theorem is referenced by: xrralrecnnle 42891 iunhoiioolem 44182 smflimlem4 44275 |
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