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Theorem cauappcvgprlemrnd 7478
 Description: Lemma for cauappcvgpr 7490. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (𝜑𝐹:QQ)
cauappcvgpr.app (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
Assertion
Ref Expression
cauappcvgprlemrnd (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐿,𝑟,𝑠   𝐴,𝑠,𝑝   𝐹,𝑙,𝑢,𝑝,𝑞,𝑟,𝑠   𝜑,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑟,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemrnd
StepHypRef Expression
1 cauappcvgpr.f . . . . . 6 (𝜑𝐹:QQ)
2 cauappcvgpr.app . . . . . 6 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
3 cauappcvgpr.bnd . . . . . 6 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
4 cauappcvgpr.lim . . . . . 6 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
51, 2, 3, 4cauappcvgprlemopl 7474 . . . . 5 ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))
65ex 114 . . . 4 (𝜑 → (𝑠 ∈ (1st𝐿) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
71, 2, 3, 4cauappcvgprlemlol 7475 . . . . . 6 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
873expib 1185 . . . . 5 (𝜑 → ((𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿)))
98rexlimdvw 2554 . . . 4 (𝜑 → (∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿)))
106, 9impbid 128 . . 3 (𝜑 → (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
1110ralrimivw 2507 . 2 (𝜑 → ∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))))
121, 2, 3, 4cauappcvgprlemopu 7476 . . . . 5 ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))
1312ex 114 . . . 4 (𝜑 → (𝑟 ∈ (2nd𝐿) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
141, 2, 3, 4cauappcvgprlemupu 7477 . . . . . 6 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
15143expib 1185 . . . . 5 (𝜑 → ((𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿)))
1615rexlimdvw 2554 . . . 4 (𝜑 → (∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿)))
1713, 16impbid 128 . . 3 (𝜑 → (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
1817ralrimivw 2507 . 2 (𝜑 → ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿))))
1911, 18jca 304 1 (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {crab 2421  ⟨cop 3531   class class class wbr 3933  ⟶wf 5123  ‘cfv 5127  (class class class)co 5778  1st c1st 6040  2nd c2nd 6041  Qcnq 7108   +Q cplq 7110
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