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Theorem cauappcvgprlemlim 7481
Description: Lemma for cauappcvgpr 7482. 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 274 . . . . 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 274 . . . . 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 274 . . . . 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 520 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑥Q)
9 simprr 521 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑦Q)
102, 4, 6, 7, 8, 9cauappcvgprlem1 7479 . . . 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 7480 . . . 4 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩)
1210, 11jca 304 . . 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 2514 . 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 5421 . . . . . . . 8 (𝑥 = 𝑞 → (𝐹𝑥) = (𝐹𝑞))
1514breq2d 3941 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q (𝐹𝑥) ↔ 𝑙 <Q (𝐹𝑞)))
1615abbidv 2257 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝐹𝑥)} = {𝑙𝑙 <Q (𝐹𝑞)})
1714breq1d 3939 . . . . . . 7 (𝑥 = 𝑞 → ((𝐹𝑥) <Q 𝑢 ↔ (𝐹𝑞) <Q 𝑢))
1817abbidv 2257 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ (𝐹𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹𝑞) <Q 𝑢})
1916, 18opeq12d 3713 . . . . 5 (𝑥 = 𝑞 → ⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩)
20 oveq1 5781 . . . . . . . . 9 (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
2120breq2d 3941 . . . . . . . 8 (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
2221abbidv 2257 . . . . . . 7 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑦)})
2320breq1d 3939 . . . . . . . 8 (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
2423abbidv 2257 . . . . . . 7 (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
2522, 24opeq12d 3713 . . . . . 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 5790 . . . . 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 3942 . . . 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 5792 . . . . . . . 8 (𝑥 = 𝑞 → ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑦)))
2928breq2d 3941 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))))
3029abbidv 2257 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))})
3128breq1d 3939 . . . . . . 7 (𝑥 = 𝑞 → (((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
3231abbidv 2257 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢})
3330, 32opeq12d 3713 . . . . 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 3941 . . . 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 464 . . 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 5782 . . . . . . . . 9 (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
3736breq2d 3941 . . . . . . . 8 (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
3837abbidv 2257 . . . . . . 7 (𝑦 = 𝑟 → {𝑙𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑟)})
3936breq1d 3939 . . . . . . . 8 (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
4039abbidv 2257 . . . . . . 7 (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
4138, 40opeq12d 3713 . . . . . 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 5790 . . . . 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 3941 . . . 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 5790 . . . . . . . 8 (𝑦 = 𝑟 → ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑟)))
4544breq2d 3941 . . . . . . 7 (𝑦 = 𝑟 → (𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))))
4645abbidv 2257 . . . . . 6 (𝑦 = 𝑟 → {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))})
4744breq1d 3939 . . . . . . 7 (𝑦 = 𝑟 → (((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
4847abbidv 2257 . . . . . 6 (𝑦 = 𝑟 → {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢})
4946, 48opeq12d 3713 . . . . 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 3941 . . . 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 464 . . 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 2665 . 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 121 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 103   = wceq 1331  wcel 1480  {cab 2125  wral 2416  wrex 2417  {crab 2420  cop 3530   class class class wbr 3929  wf 5119  cfv 5123  (class class class)co 5774  Qcnq 7100   +Q cplq 7102   <Q cltq 7105   +P cpp 7113  <P cltp 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288  df-iltp 7290
This theorem is referenced by:  cauappcvgpr  7482
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