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Theorem caucvgprprlemk 7503
 Description: Lemma for caucvgprpr 7532. 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 7144 . . . . . 6 <N ⊆ (N × N)
32brel 4591 . . . . 5 (𝐽 <N 𝐾 → (𝐽N𝐾N))
4 ltnnnq 7243 . . . . 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 146 . . 3 (𝜑 → [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q )
7 ltrnqi 7241 . . 3 ([⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))
8 ltnqpri 7414 . . 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 7416 . . 3 <P Or P
12 ltrelpr 7325 . . 3 <P ⊆ (P × P)
1311, 12sotri 4934 . 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 408 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 103   ↔ wb 104   ∈ wcel 1480  {cab 2125  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  1oc1o 6306  [cec 6427  Ncnpi 7092
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