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Theorem archrecnq 6761
 Description: Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
Assertion
Ref Expression
archrecnq (𝐴Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴)
Distinct variable group:   𝐴,𝑗

Proof of Theorem archrecnq
StepHypRef Expression
1 recclnq 6490 . . 3 (𝐴Q → (*Q𝐴) ∈ Q)
2 archnqq 6515 . . 3 ((*Q𝐴) ∈ Q → ∃𝑗N (*Q𝐴) <Q [⟨𝑗, 1𝑜⟩] ~Q )
31, 2syl 14 . 2 (𝐴Q → ∃𝑗N (*Q𝐴) <Q [⟨𝑗, 1𝑜⟩] ~Q )
4 nnnq 6520 . . . . 5 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
5 ltrnqg 6518 . . . . 5 (((*Q𝐴) ∈ Q ∧ [⟨𝑗, 1𝑜⟩] ~QQ) → ((*Q𝐴) <Q [⟨𝑗, 1𝑜⟩] ~Q ↔ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q (*Q‘(*Q𝐴))))
61, 4, 5syl2an 273 . . . 4 ((𝐴Q𝑗N) → ((*Q𝐴) <Q [⟨𝑗, 1𝑜⟩] ~Q ↔ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q (*Q‘(*Q𝐴))))
7 recrecnq 6492 . . . . . 6 (𝐴Q → (*Q‘(*Q𝐴)) = 𝐴)
87breq2d 3776 . . . . 5 (𝐴Q → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q (*Q‘(*Q𝐴)) ↔ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴))
98adantr 261 . . . 4 ((𝐴Q𝑗N) → ((*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q (*Q‘(*Q𝐴)) ↔ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴))
106, 9bitrd 177 . . 3 ((𝐴Q𝑗N) → ((*Q𝐴) <Q [⟨𝑗, 1𝑜⟩] ~Q ↔ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴))
1110rexbidva 2323 . 2 (𝐴Q → (∃𝑗N (*Q𝐴) <Q [⟨𝑗, 1𝑜⟩] ~Q ↔ ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴))
123, 11mpbid 135 1 (𝐴Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  ∃wrex 2307  ⟨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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