Theorem cauappcvgprlemlim 7469

Description: Lemma for cauappcvgpr 7470. 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 274	. . . . 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 274	. . . . 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 274	. . . . 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 520	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑥 ∈ Q)
9		simprr 521	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑦 ∈ Q)
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7467	. . . 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 7468	. . . 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 304	. . 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 2514	. 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 5421	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝐹‘𝑥) = (𝐹‘𝑞))
15	14	breq2d 3941	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝐹‘𝑥) ↔ 𝑙 <Q (𝐹‘𝑞)))
16	15	abbidv 2257	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)} = {𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)})
17	14	breq1d 3939	. . . . . . 7 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) <Q 𝑢 ↔ (𝐹‘𝑞) <Q 𝑢))
18	17	abbidv 2257	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢})
19	16, 18	opeq12d 3713	. . . . 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 5781	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
21	20	breq2d 3941	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
22	21	abbidv 2257	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)})
23	20	breq1d 3939	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
24	23	abbidv 2257	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
25	22, 24	opeq12d 3713	. . . . . 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 5790	. . . . 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 3942	. . . 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 5792	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)))
29	28	breq2d 3941	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))))
30	29	abbidv 2257	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))})
31	28	breq1d 3939	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
32	31	abbidv 2257	. . . . . 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 3713	. . . . 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 3941	. . . 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 464	. . 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 5782	. . . . . . . . 9 ⊢ (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
37	36	breq2d 3941	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
38	37	abbidv 2257	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)})
39	36	breq1d 3939	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
40	39	abbidv 2257	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
41	38, 40	opeq12d 3713	. . . . . 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 5790	. . . . 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 3941	. . . 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 5790	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)))
45	44	breq2d 3941	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))))
46	45	abbidv 2257	. . . . . 6 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))})
47	44	breq1d 3939	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
48	47	abbidv 2257	. . . . . 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 3713	. . . . 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 3941	. . . 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 464	. . 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 2665	. 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 121	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 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 7088   +_Q cplq 7090   <_Q cltq 7093   +_P cpp 7101  <_P cltp 7103

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 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  cauappcvgpr  7470