Theorem cauappcvgprlemlim 7623

Description: Lemma for cauappcvgpr 7624. 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 526	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑥 ∈ Q)
9		simprr 527	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑦 ∈ Q)
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7621	. . . 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 7622	. . . 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 2552	. 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 5496	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝐹‘𝑥) = (𝐹‘𝑞))
15	14	breq2d 4001	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝐹‘𝑥) ↔ 𝑙 <Q (𝐹‘𝑞)))
16	15	abbidv 2288	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)} = {𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)})
17	14	breq1d 3999	. . . . . . 7 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) <Q 𝑢 ↔ (𝐹‘𝑞) <Q 𝑢))
18	17	abbidv 2288	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢})
19	16, 18	opeq12d 3773	. . . . 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 5860	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
21	20	breq2d 4001	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
22	21	abbidv 2288	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)})
23	20	breq1d 3999	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
24	23	abbidv 2288	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
25	22, 24	opeq12d 3773	. . . . . 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 5869	. . . . 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 4002	. . . 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 5871	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)))
29	28	breq2d 4001	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))))
30	29	abbidv 2288	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))})
31	28	breq1d 3999	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
32	31	abbidv 2288	. . . . . 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 3773	. . . . 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 4001	. . . 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 470	. . 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 5861	. . . . . . . . 9 ⊢ (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
37	36	breq2d 4001	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
38	37	abbidv 2288	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)})
39	36	breq1d 3999	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
40	39	abbidv 2288	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
41	38, 40	opeq12d 3773	. . . . . 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 5869	. . . . 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 4001	. . . 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 5869	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)))
45	44	breq2d 4001	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))))
46	45	abbidv 2288	. . . . . 6 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))})
47	44	breq1d 3999	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
48	47	abbidv 2288	. . . . . 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 3773	. . . . 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 4001	. . . 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 470	. . 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 2709	. 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 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  {crab 2452  ⟨cop 3586   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  Qcnq 7242   +_Q cplq 7244   <_Q cltq 7247   +_P cpp 7255  <_P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by:  cauappcvgpr  7624