Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemupu GIF version

Theorem caucvgprprlemupu 7530
 Description: Lemma for caucvgprpr 7542. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
Assertion
Ref Expression
caucvgprprlemupu ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → 𝑡 ∈ (2nd𝐿))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹,𝑟,𝑠   𝐿,𝑠   𝑝,𝑙,𝑞,𝑡,𝑟,𝑠   𝑢,𝑝,𝑞,𝑡   𝜑,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑡,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑞,𝑝)   𝐿(𝑢,𝑡,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemupu
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 ltrelnq 7195 . . . . 5 <Q ⊆ (Q × Q)
21brel 4597 . . . 4 (𝑠 <Q 𝑡 → (𝑠Q𝑡Q))
32simprd 113 . . 3 (𝑠 <Q 𝑡𝑡Q)
433ad2ant2 1004 . 2 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → 𝑡Q)
5 caucvgprpr.lim . . . . . 6 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
65caucvgprprlemelu 7516 . . . . 5 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
76simprbi 273 . . . 4 (𝑠 ∈ (2nd𝐿) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩)
873ad2ant3 1005 . . 3 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩)
9 ltnqpri 7424 . . . . . 6 (𝑠 <Q 𝑡 → ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
1093ad2ant2 1004 . . . . 5 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
11 ltsopr 7426 . . . . . . 7 <P Or P
12 ltrelpr 7335 . . . . . . 7 <P ⊆ (P × P)
1311, 12sotri 4940 . . . . . 6 ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
1413expcom 115 . . . . 5 (⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
1510, 14syl 14 . . . 4 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
1615reximdv 2536 . . 3 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → (∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
178, 16mpd 13 . 2 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
185caucvgprprlemelu 7516 . 2 (𝑡 ∈ (2nd𝐿) ↔ (𝑡Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
194, 17, 18sylanbrc 414 1 ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → 𝑡 ∈ (2nd𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  {crab 2421  ⟨cop 3533   class class class wbr 3935  ⟶wf 5125  ‘cfv 5129  (class class class)co 5780  2nd c2nd 6043  1oc1o 6312  [cec 6433  Ncnpi 7102
 Copyright terms: Public domain W3C validator