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Theorem caucvgprlemopu 6769
 Description: Lemma for caucvgpr 6780. The upper cut of the putative limit is open. (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
caucvgprlemopu ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (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 caucvgprlemopu
StepHypRef Expression
1 breq2 3768 . . . . . 6 (𝑢 = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
21rexbidv 2327 . . . . 5 (𝑢 = 𝑟 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
3 caucvgpr.lim . . . . . . 7 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
43fveq2i 5181 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
5 nqex 6461 . . . . . . . 8 Q ∈ V
65rabex 3901 . . . . . . 7 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
75rabex 3901 . . . . . . 7 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
86, 7op2nd 5774 . . . . . 6 (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}
94, 8eqtri 2060 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
102, 9elrab2 2700 . . . 4 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
1110simprbi 260 . . 3 (𝑟 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
1211adantl 262 . 2 ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
13 simprr 484 . . . 4 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)
14 ltbtwnnqq 6513 . . . 4 (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟 ↔ ∃𝑠Q (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟))
1513, 14sylib 127 . . 3 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → ∃𝑠Q (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟))
16 simprr 484 . . . . . 6 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑠 <Q 𝑟)
17 simplr 482 . . . . . . 7 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑠Q)
18 simplrl 487 . . . . . . . . 9 ((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) → 𝑗N)
1918adantr 261 . . . . . . . 8 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑗N)
20 simprl 483 . . . . . . . 8 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
21 rspe 2370 . . . . . . . 8 ((𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
2219, 20, 21syl2anc 391 . . . . . . 7 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
23 breq2 3768 . . . . . . . . 9 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2423rexbidv 2327 . . . . . . . 8 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2524, 9elrab2 2700 . . . . . . 7 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2617, 22, 25sylanbrc 394 . . . . . 6 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → 𝑠 ∈ (2nd𝐿))
2716, 26jca 290 . . . . 5 (((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) ∧ (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟)) → (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
2827ex 108 . . . 4 ((((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) ∧ 𝑠Q) → ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟) → (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
2928reximdva 2421 . . 3 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → (∃𝑠Q (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
3015, 29mpd 13 . 2 (((𝜑𝑟 ∈ (2nd𝐿)) ∧ (𝑗N ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
3112, 30rexlimddv 2437 1 ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  2nd c2nd 5766  1𝑜c1o 5994  [cec 6104  Ncnpi 6370
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