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Theorem caucvgprlemupu 7499
 Description: Lemma for caucvgpr 7509. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemupu ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
Distinct variable groups:   𝐴,𝑗   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹   𝑗,𝐿,𝑟,𝑠   𝑗,𝑙,𝑠   𝜑,𝑗,𝑟,𝑠   𝑢,𝑗,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑗,𝑘,𝑛)   𝐿(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprlemupu
StepHypRef Expression
1 ltrelnq 7192 . . . . 5 <Q ⊆ (Q × Q)
21brel 4594 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simprd 113 . . 3 (𝑠 <Q 𝑟𝑟Q)
433ad2ant2 1003 . 2 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟Q)
5 breq2 3936 . . . . . . 7 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
65rexbidv 2438 . . . . . 6 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
7 caucvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
87fveq2i 5427 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
9 nqex 7190 . . . . . . . . 9 Q ∈ V
109rabex 4075 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
119rabex 4075 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
1210, 11op2nd 6048 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
138, 12eqtri 2160 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
146, 13elrab2 2843 . . . . 5 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
1514simprbi 273 . . . 4 (𝑠 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
16153ad2ant3 1004 . . 3 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
17 ltsonq 7225 . . . . . . 7 <Q Or Q
1817, 1sotri 4937 . . . . . 6 ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟) → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
1918expcom 115 . . . . 5 (𝑠 <Q 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
20193ad2ant2 1003 . . . 4 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2120reximdv 2533 . . 3 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2216, 21mpd 13 . 2 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
23 breq2 3936 . . . 4 (𝑢 = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2423rexbidv 2438 . . 3 (𝑢 = 𝑟 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2524, 13elrab2 2843 . 2 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
264, 22, 25sylanbrc 413 1 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3932  ⟶wf 5122  ‘cfv 5126  (class class class)co 5777  2nd c2nd 6040  1oc1o 6309  [cec 6430  Ncnpi 7099
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