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Theorem recnnpr 6646
 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 6520 . 2 (𝐴N → [⟨𝐴, 1𝑜⟩] ~QQ)
2 recclnq 6490 . 2 ([⟨𝐴, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) ∈ Q)
3 nqprlu 6645 . 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 1393  {cab 2026  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  1𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~Q ceq 6377  Qcnq 6378  *Qcrq 6382
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