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Theorem cauappcvgprlemlim 7651
Description: Lemma for cauappcvgpr 7652. 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 276 . . . . 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 276 . . . . 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 276 . . . . 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 529 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑥Q)
9 simprr 531 . . . . 5 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝑦Q)
102, 4, 6, 7, 8, 9cauappcvgprlem1 7649 . . . 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 7650 . . . 4 ((𝜑 ∧ (𝑥Q𝑦Q)) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩)
1210, 11jca 306 . . 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 2559 . 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 5511 . . . . . . . 8 (𝑥 = 𝑞 → (𝐹𝑥) = (𝐹𝑞))
1514breq2d 4012 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q (𝐹𝑥) ↔ 𝑙 <Q (𝐹𝑞)))
1615abbidv 2295 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝐹𝑥)} = {𝑙𝑙 <Q (𝐹𝑞)})
1714breq1d 4010 . . . . . . 7 (𝑥 = 𝑞 → ((𝐹𝑥) <Q 𝑢 ↔ (𝐹𝑞) <Q 𝑢))
1817abbidv 2295 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ (𝐹𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹𝑞) <Q 𝑢})
1916, 18opeq12d 3784 . . . . 5 (𝑥 = 𝑞 → ⟨{𝑙𝑙 <Q (𝐹𝑥)}, {𝑢 ∣ (𝐹𝑥) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩)
20 oveq1 5876 . . . . . . . . 9 (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
2120breq2d 4012 . . . . . . . 8 (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
2221abbidv 2295 . . . . . . 7 (𝑥 = 𝑞 → {𝑙𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑦)})
2320breq1d 4010 . . . . . . . 8 (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
2423abbidv 2295 . . . . . . 7 (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
2522, 24opeq12d 3784 . . . . . 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 5885 . . . . 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 4013 . . . 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 5887 . . . . . . . 8 (𝑥 = 𝑞 → ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑦)))
2928breq2d 4012 . . . . . . 7 (𝑥 = 𝑞 → (𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))))
3029abbidv 2295 . . . . . 6 (𝑥 = 𝑞 → {𝑙𝑙 <Q ((𝐹𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))})
3128breq1d 4010 . . . . . . 7 (𝑥 = 𝑞 → (((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
3231abbidv 2295 . . . . . 6 (𝑥 = 𝑞 → {𝑢 ∣ ((𝐹𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢})
3330, 32opeq12d 3784 . . . . 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 4012 . . . 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 473 . . 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 5877 . . . . . . . . 9 (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
3736breq2d 4012 . . . . . . . 8 (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
3837abbidv 2295 . . . . . . 7 (𝑦 = 𝑟 → {𝑙𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙𝑙 <Q (𝑞 +Q 𝑟)})
3936breq1d 4010 . . . . . . . 8 (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
4039abbidv 2295 . . . . . . 7 (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
4138, 40opeq12d 3784 . . . . . 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 5885 . . . . 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 4012 . . . 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 5885 . . . . . . . 8 (𝑦 = 𝑟 → ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹𝑞) +Q (𝑞 +Q 𝑟)))
4544breq2d 4012 . . . . . . 7 (𝑦 = 𝑟 → (𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))))
4645abbidv 2295 . . . . . 6 (𝑦 = 𝑟 → {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))})
4744breq1d 4010 . . . . . . 7 (𝑦 = 𝑟 → (((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
4847abbidv 2295 . . . . . 6 (𝑦 = 𝑟 → {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢})
4946, 48opeq12d 3784 . . . . 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 4012 . . . 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 473 . . 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 2716 . 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 122 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 104   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  cop 3594   class class class wbr 4000  wf 5208  cfv 5212  (class class class)co 5869  Qcnq 7270   +Q cplq 7272   <Q cltq 7275   +P cpp 7283  <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460
This theorem is referenced by:  cauappcvgpr  7652
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