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Theorem caucvgprprlemk 6781
 Description: Lemma for caucvgprpr 6810. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
Hypotheses
Ref Expression
caucvgprprlemk.jk (𝜑𝐽 <N 𝐾)
caucvgprprlemk.jkq (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
Assertion
Ref Expression
caucvgprprlemk (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
Distinct variable groups:   𝐽,𝑙   𝑢,𝐽   𝐾,𝑙   𝑢,𝐾
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝑄(𝑢,𝑙)

Proof of Theorem caucvgprprlemk
StepHypRef Expression
1 caucvgprprlemk.jk . . . 4 (𝜑𝐽 <N 𝐾)
2 ltrelpi 6422 . . . . . 6 <N ⊆ (N × N)
32brel 4392 . . . . 5 (𝐽 <N 𝐾 → (𝐽N𝐾N))
4 ltnnnq 6521 . . . . 5 ((𝐽N𝐾N) → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q ))
51, 3, 43syl 17 . . . 4 (𝜑 → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q ))
61, 5mpbid 135 . . 3 (𝜑 → [⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q )
7 ltrnqi 6519 . . 3 ([⟨𝐽, 1𝑜⟩] ~Q <Q [⟨𝐾, 1𝑜⟩] ~Q → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
8 ltnqpri 6692 . . 3 ((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
96, 7, 83syl 17 . 2 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
10 caucvgprprlemk.jkq . 2 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
11 ltsopr 6694 . . 3 <P Or P
12 ltrelpr 6603 . . 3 <P ⊆ (P × P)
1311, 12sotri 4720 . 2 ((⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄) → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
149, 10, 13syl2anc 391 1 (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  {cab 2026  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  1𝑜c1o 5994  [cec 6104  Ncnpi 6370
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