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Theorem recnnpr 7320
 Description: The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
Assertion
Ref Expression
recnnpr (𝐴N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
Distinct variable group:   𝐴,𝑙,𝑢

Proof of Theorem recnnpr
StepHypRef Expression
1 nnnq 7194 . 2 (𝐴N → [⟨𝐴, 1o⟩] ~QQ)
2 recclnq 7164 . 2 ([⟨𝐴, 1o⟩] ~QQ → (*Q‘[⟨𝐴, 1o⟩] ~Q ) ∈ Q)
3 nqprlu 7319 . 2 ((*Q‘[⟨𝐴, 1o⟩] ~Q ) ∈ Q → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
41, 2, 33syl 17 1 (𝐴N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1463  {cab 2101  ⟨cop 3498   class class class wbr 3897  ‘cfv 5091  1oc1o 6272  [cec 6393  Ncnpi 7044   ~Q ceq 7051  Qcnq 7052  *Qcrq 7056
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