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	⊢ (𝜑 → 𝐹:Q⟶Q)
cauappcvgpr.app	⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd	⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
cauappcvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩

Assertion

Ref	Expression
cauappcvgprlemlim	⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +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.

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:Q⟶Q)
2	1	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝐹:Q⟶Q)
3		cauappcvgpr.app	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
4	3	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
5		cauappcvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
6	5	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
7		cauappcvgpr.lim	. . . . 5 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
8		simprl 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑥 ∈ Q)
9		simprr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑦 ∈ Q)
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 6757	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩))
11	2, 4, 6, 7, 8, 9	cauappcvgprlem2 6758	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩)
12	10, 11	jca 290	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩))
13	12	ralrimivva 2401	. 2 ⊢ (𝜑 → ∀𝑥 ∈ Q ∀𝑦 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩))
14		fveq2 5178	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝐹‘𝑥) = (𝐹‘𝑞))
15	14	breq2d 3776	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝐹‘𝑥) ↔ 𝑙 <Q (𝐹‘𝑞)))
16	15	abbidv 2155	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)} = {𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)})
17	14	breq1d 3774	. . . . . . 7 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) <Q 𝑢 ↔ (𝐹‘𝑞) <Q 𝑢))
18	17	abbidv 2155	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢})
19	16, 18	opeq12d 3557	. . . . 5 ⊢ (𝑥 = 𝑞 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩)
20		oveq1 5519	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
21	20	breq2d 3776	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
22	21	abbidv 2155	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)})
23	20	breq1d 3774	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
24	23	abbidv 2155	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
25	22, 24	opeq12d 3557	. . . . . 6 ⊢ (𝑥 = 𝑞 → ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩)
26	25	oveq2d 5528	. . . . 5 ⊢ (𝑥 = 𝑞 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) = (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩))
27	19, 26	breq12d 3777	. . . 4 ⊢ (𝑥 = 𝑞 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩)))
28	14, 20	oveq12d 5530	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)))
29	28	breq2d 3776	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))))
30	29	abbidv 2155	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))})
31	28	breq1d 3774	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
32	31	abbidv 2155	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢})
33	30, 32	opeq12d 3557	. . . . 5 ⊢ (𝑥 = 𝑞 → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩)
34	33	breq2d 3776	. . . 4 ⊢ (𝑥 = 𝑞 → (𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩ ↔ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩))
35	27, 34	anbi12d 442	. . 3 ⊢ (𝑥 = 𝑞 → ((⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩)))
36		oveq2 5520	. . . . . . . . 9 ⊢ (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
37	36	breq2d 3776	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
38	37	abbidv 2155	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)})
39	36	breq1d 3774	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
40	39	abbidv 2155	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
41	38, 40	opeq12d 3557	. . . . . 6 ⊢ (𝑦 = 𝑟 → ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩)
42	41	oveq2d 5528	. . . . 5 ⊢ (𝑦 = 𝑟 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) = (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩))
43	42	breq2d 3776	. . . 4 ⊢ (𝑦 = 𝑟 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩)))
44	36	oveq2d 5528	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)))
45	44	breq2d 3776	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))))
46	45	abbidv 2155	. . . . . 6 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))})
47	44	breq1d 3774	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
48	47	abbidv 2155	. . . . . 6 ⊢ (𝑦 = 𝑟 → {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢} = {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢})
49	46, 48	opeq12d 3557	. . . . 5 ⊢ (𝑦 = 𝑟 → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)
50	49	breq2d 3776	. . . 4 ⊢ (𝑦 = 𝑟 → (𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩ ↔ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
51	43, 50	anbi12d 442	. . 3 ⊢ (𝑦 = 𝑟 → ((⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)}, {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)))
52	35, 51	cbvral2v 2541	. 2 ⊢ (∀𝑥 ∈ Q ∀𝑦 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)}, {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)}, {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))}, {𝑢 ∣ ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢}⟩) ↔ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
53	13, 52	sylib 127	1 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))

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   <_Q cltq 6383   +_P cpp 6391  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568

This theorem is referenced by:  cauappcvgpr  6760