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Theorem cauappcvgprlemlim 6759
 Description: Lemma for cauappcvgpr 6760. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-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
cauappcvgprlemlim (𝜑 → ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑙,𝑝,𝑞,𝑟,𝑢   𝐿,𝑟
Allowed substitution hints:   𝜑(𝑢,𝑟,𝑙)   𝐴(𝑢,𝑟,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemlim
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . 6 (𝜑𝐹:QQ)
21adantr 261 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝐹:QQ)
3 cauappcvgpr.app . . . . . 6 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
43adantr 261 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
5 cauappcvgpr.bnd . . . . . 6 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
65adantr 261 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
7 cauappcvgpr.lim . . . . 5 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
8 simprl 483 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑥Q)
9 simprr 484 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑦Q)
102, 4, 6, 7, 8, 9cauappcvgprlem1 6757 . . . 4 ((𝜑 ∧ (𝑥Q𝑦Q)) → ⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩))
112, 4, 6, 7, 8, 9cauappcvgprlem2 6758 . . . 4 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩)
1210, 11jca 290 . . 3 ((𝜑 ∧ (𝑥Q𝑦Q)) → (⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩))
1312ralrimivva 2401 . 2 (𝜑 → ∀𝑥Q𝑦Q (⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩))
14 fveq2 5178 . . . . . . . 8 (𝑥 = 𝑞 → (𝐹𝑥) = (𝐹𝑞))
1514breq2d 3776 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q (𝐹𝑥) ↔ 𝑙 <Q (𝐹𝑞)))
1615abbidv 2155 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝐹𝑥)} = {𝑙𝑙 <Q (𝐹𝑞)})
1714breq1d 3774 . . . . . . 7 (𝑥 = 𝑞 → ((𝐹𝑥) <Q 𝑢 ↔ (𝐹𝑞) <Q 𝑢))
1817abbidv 2155 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ (𝐹𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹𝑞) <Q 𝑢})
1916, 18opeq12d 3557 . . . . 5 (𝑥 = 𝑞 → ⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩)
20 oveq1 5519 . . . . . . . . 9 (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
2120breq2d 3776 . . . . . . . 8 (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
2221abbidv 2155 . . . . . . 7 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑦)})
2320breq1d 3774 . . . . . . . 8 (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
2423abbidv 2155 . . . . . . 7 (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
2522, 24opeq12d 3557 . . . . . 6 (𝑥 = 𝑞 → ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩)
2625oveq2d 5528 . . . . 5 (𝑥 = 𝑞 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) = (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩))
2719, 26breq12d 3777 . . . 4 (𝑥 = 𝑞 → (⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩)))
2814, 20oveq12d 5530 . . . . . . . 8 (𝑥 = 𝑞 → ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑦)))
2928breq2d 3776 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))))
3029abbidv 2155 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))})
3128breq1d 3774 . . . . . . 7 (𝑥 = 𝑞 → (((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
3231abbidv 2155 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢})
3330, 32opeq12d 3557 . . . . 5 (𝑥 = 𝑞 → ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩)
3433breq2d 3776 . . . 4 (𝑥 = 𝑞 → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩ ↔ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩))
3527, 34anbi12d 442 . . 3 (𝑥 = 𝑞 → ((⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩)))
36 oveq2 5520 . . . . . . . . 9 (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
3736breq2d 3776 . . . . . . . 8 (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
3837abbidv 2155 . . . . . . 7 (𝑦 = 𝑟 → {𝑙𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑟)})
3936breq1d 3774 . . . . . . . 8 (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
4039abbidv 2155 . . . . . . 7 (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
4138, 40opeq12d 3557 . . . . . 6 (𝑦 = 𝑟 → ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩)
4241oveq2d 5528 . . . . 5 (𝑦 = 𝑟 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) = (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩))
4342breq2d 3776 . . . 4 (𝑦 = 𝑟 → (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩)))
4436oveq2d 5528 . . . . . . . 8 (𝑦 = 𝑟 → ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑟)))
4544breq2d 3776 . . . . . . 7 (𝑦 = 𝑟 → (𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))))
4645abbidv 2155 . . . . . 6 (𝑦 = 𝑟 → {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))})
4744breq1d 3774 . . . . . . 7 (𝑦 = 𝑟 → (((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
4847abbidv 2155 . . . . . 6 (𝑦 = 𝑟 → {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢})
4946, 48opeq12d 3557 . . . . 5 (𝑦 = 𝑟 → ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)
5049breq2d 3776 . . . 4 (𝑦 = 𝑟 → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩ ↔ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
5143, 50anbi12d 442 . . 3 (𝑦 = 𝑟 → ((⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)))
5235, 51cbvral2v 2541 . 2 (∀𝑥Q𝑦Q (⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩) ↔ ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
5313, 52sylib 127 1 (𝜑 → ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  Qcnq 6378   +Q cplq 6380
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