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 12409 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) | |
2 | 1 | rpred 12425 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ) |
3 | 1 | rpge0d 12429 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (1 / 𝐴)) |
4 | flge0nn0 13184 | . . . 4 ⊢ (((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)) → (⌊‘(1 / 𝐴)) ∈ ℕ0) | |
5 | 2, 3, 4 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (⌊‘(1 / 𝐴)) ∈ ℕ0) |
6 | nn0p1nn 11930 | . . 3 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℕ0 → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ) |
8 | flltp1 13164 | . . . . 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 12423 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ+) |
11 | 1, 10 | ltrecd 12443 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((1 / 𝐴) < ((⌊‘(1 / 𝐴)) + 1) ↔ (1 / ((⌊‘(1 / 𝐴)) + 1)) < (1 / (1 / 𝐴)))) |
12 | 9, 11 | mpbid 234 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (1 / ((⌊‘(1 / 𝐴)) + 1)) < (1 / (1 / 𝐴))) |
13 | rpcn 12393 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
14 | rpne0 12399 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
15 | 13, 14 | recrecd 11407 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (1 / (1 / 𝐴)) = 𝐴) |
16 | 12, 15 | breqtrd 5085 | . 2 ⊢ (𝐴 ∈ ℝ+ → (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴) |
17 | oveq2 7158 | . . . 4 ⊢ (𝑛 = ((⌊‘(1 / 𝐴)) + 1) → (1 / 𝑛) = (1 / ((⌊‘(1 / 𝐴)) + 1))) | |
18 | 17 | breq1d 5069 | . . 3 ⊢ (𝑛 = ((⌊‘(1 / 𝐴)) + 1) → ((1 / 𝑛) < 𝐴 ↔ (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴)) |
19 | 18 | rspcev 3623 | . 2 ⊢ ((((⌊‘(1 / 𝐴)) + 1) ∈ ℕ ∧ (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴) |
20 | 7, 16, 19 | syl2anc 586 | 1 ⊢ (𝐴 ∈ ℝ+ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 / cdiv 11291 ℕcn 11632 ℕ0cn0 11891 ℝ+crp 12383 ⌊cfl 13154 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 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-inf 8901 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-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13156 |
This theorem is referenced by: xrralrecnnle 41645 iunhoiioolem 42950 smflimlem4 43043 |
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