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Theorem cauappcvgprlemlim 6816
Description: Lemma for cauappcvgpr 6817. 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 265 . . . . 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 265 . . . . 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 265 . . . . 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 491 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑥Q)
9 simprr 492 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑦Q)
102, 4, 6, 7, 8, 9cauappcvgprlem1 6814 . . . 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 6815 . . . 4 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩)
1210, 11jca 294 . . 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 2418 . 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 5205 . . . . . . . 8 (𝑥 = 𝑞 → (𝐹𝑥) = (𝐹𝑞))
1514breq2d 3803 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q (𝐹𝑥) ↔ 𝑙 <Q (𝐹𝑞)))
1615abbidv 2171 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝐹𝑥)} = {𝑙𝑙 <Q (𝐹𝑞)})
1714breq1d 3801 . . . . . . 7 (𝑥 = 𝑞 → ((𝐹𝑥) <Q 𝑢 ↔ (𝐹𝑞) <Q 𝑢))
1817abbidv 2171 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ (𝐹𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹𝑞) <Q 𝑢})
1916, 18opeq12d 3584 . . . . 5 (𝑥 = 𝑞 → ⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩)
20 oveq1 5546 . . . . . . . . 9 (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
2120breq2d 3803 . . . . . . . 8 (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
2221abbidv 2171 . . . . . . 7 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑦)})
2320breq1d 3801 . . . . . . . 8 (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
2423abbidv 2171 . . . . . . 7 (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
2522, 24opeq12d 3584 . . . . . 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 5555 . . . . 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 3804 . . . 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 5557 . . . . . . . 8 (𝑥 = 𝑞 → ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑦)))
2928breq2d 3803 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))))
3029abbidv 2171 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))})
3128breq1d 3801 . . . . . . 7 (𝑥 = 𝑞 → (((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
3231abbidv 2171 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢})
3330, 32opeq12d 3584 . . . . 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 3803 . . . 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 450 . . 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 5547 . . . . . . . . 9 (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
3736breq2d 3803 . . . . . . . 8 (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
3837abbidv 2171 . . . . . . 7 (𝑦 = 𝑟 → {𝑙𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑟)})
3936breq1d 3801 . . . . . . . 8 (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
4039abbidv 2171 . . . . . . 7 (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
4138, 40opeq12d 3584 . . . . . 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 5555 . . . . 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 3803 . . . 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 5555 . . . . . . . 8 (𝑦 = 𝑟 → ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑟)))
4544breq2d 3803 . . . . . . 7 (𝑦 = 𝑟 → (𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))))
4645abbidv 2171 . . . . . 6 (𝑦 = 𝑟 → {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))})
4744breq1d 3801 . . . . . . 7 (𝑦 = 𝑟 → (((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
4847abbidv 2171 . . . . . 6 (𝑦 = 𝑟 → {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢})
4946, 48opeq12d 3584 . . . . 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 3803 . . . 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 450 . . 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 2558 . 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 131 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 101   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  Qcnq 6435   +Q cplq 6437   <Q cltq 6440   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-iltp 6625
This theorem is referenced by:  cauappcvgpr  6817
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