Theorem cauappcvgprlemlim 7721

Description: Lemma for cauappcvgpr 7722. 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 276	. . . . 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 276	. . . . 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 276	. . . . 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 529	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑥 ∈ Q)
9		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑦 ∈ Q)
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7719	. . . 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 7720	. . . 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 306	. . 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 2576	. 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 5554	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝐹‘𝑥) = (𝐹‘𝑞))
15	14	breq2d 4041	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝐹‘𝑥) ↔ 𝑙 <Q (𝐹‘𝑞)))
16	15	abbidv 2311	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)} = {𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)})
17	14	breq1d 4039	. . . . . . 7 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) <Q 𝑢 ↔ (𝐹‘𝑞) <Q 𝑢))
18	17	abbidv 2311	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢})
19	16, 18	opeq12d 3812	. . . . 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 5925	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
21	20	breq2d 4041	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
22	21	abbidv 2311	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)})
23	20	breq1d 4039	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
24	23	abbidv 2311	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
25	22, 24	opeq12d 3812	. . . . . 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 5934	. . . . 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 4042	. . . 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 5936	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)))
29	28	breq2d 4041	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))))
30	29	abbidv 2311	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))})
31	28	breq1d 4039	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
32	31	abbidv 2311	. . . . . 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 3812	. . . . 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 4041	. . . 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 473	. . 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 5926	. . . . . . . . 9 ⊢ (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
37	36	breq2d 4041	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
38	37	abbidv 2311	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)})
39	36	breq1d 4039	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
40	39	abbidv 2311	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
41	38, 40	opeq12d 3812	. . . . . 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 5934	. . . . 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 4041	. . . 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 5934	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)))
45	44	breq2d 4041	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))))
46	45	abbidv 2311	. . . . . 6 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))})
47	44	breq1d 4039	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
48	47	abbidv 2311	. . . . . 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 3812	. . . . 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 4041	. . . 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 473	. . 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 2739	. 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 122	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 104   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  {crab 2476  ⟨cop 3621   class class class wbr 4029  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  Qcnq 7340   +_Q cplq 7342   <_Q cltq 7345   +_P cpp 7353  <_P cltp 7355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  cauappcvgpr  7722