rpgtrecnn - Mathbox for Glauco Siliprandi

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Theorem rpgtrecnn 38538

Description: Any positive real number is greater than the reciprocal of a positive integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Assertion**
Ref	Expression
rpgtrecnn	⊢ (𝐴 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)

Distinct variable group: 𝐴,𝑛

**Proof of Theorem rpgtrecnn**
Step	Hyp	Ref	Expression
1		rpreccl 11733	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
2	1	rpred 11748	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ)
3	1	rpge0d 11752	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ (1 / 𝐴))
4		flge0nn0 12483	. . . 4 ⊢ (((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)) → (⌊‘(1 / 𝐴)) ∈ ℕ₀)
5	2, 3, 4	syl2anc 691	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘(1 / 𝐴)) ∈ ℕ₀)
6		nn0p1nn 11209	. . 3 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℕ₀ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
7	5, 6	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
8		flltp1 12463	. . . . 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 11746	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ⁺)
11	1, 10	ltrecd 11766	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((1 / 𝐴) < ((⌊‘(1 / 𝐴)) + 1) ↔ (1 / ((⌊‘(1 / 𝐴)) + 1)) < (1 / (1 / 𝐴))))
12	9, 11	mpbid 221	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘(1 / 𝐴)) + 1)) < (1 / (1 / 𝐴)))
13		rpcn 11717	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
14		rpne0 11724	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
15	13, 14	recrecd 10677	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 / 𝐴)) = 𝐴)
16	12, 15	breqtrd 4609	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴)
17		oveq2 6557	. . . 4 ⊢ (𝑛 = ((⌊‘(1 / 𝐴)) + 1) → (1 / 𝑛) = (1 / ((⌊‘(1 / 𝐴)) + 1)))
18	17	breq1d 4593	. . 3 ⊢ (𝑛 = ((⌊‘(1 / 𝐴)) + 1) → ((1 / 𝑛) < 𝐴 ↔ (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴))
19	18	rspcev 3282	. 2 ⊢ ((((⌊‘(1 / 𝐴)) + 1) ∈ ℕ ∧ (1 / ((⌊‘(1 / 𝐴)) + 1)) < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)
20	7, 16, 19	syl2anc 691	1 ⊢ (𝐴 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℝ⁺crp 11708 ⌊cfl 12453

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455

This theorem is referenced by: xrralrecnnle 38543 iunhoiioolem 39566 smflimlem4 39660

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