Theorem cauappcvgprlemlim 7370

Description: Lemma for cauappcvgpr 7371. 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 272	. . . . 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 272	. . . . 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 272	. . . . 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 501	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑥 ∈ Q)
9		simprr 502	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → 𝑦 ∈ Q)
10	2, 4, 6, 7, 8, 9	cauappcvgprlem1 7368	. . . 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 7369	. . . 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 302	. . 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 2473	. 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 5353	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝐹‘𝑥) = (𝐹‘𝑞))
15	14	breq2d 3887	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝐹‘𝑥) ↔ 𝑙 <Q (𝐹‘𝑞)))
16	15	abbidv 2217	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝐹‘𝑥)} = {𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)})
17	14	breq1d 3885	. . . . . . 7 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) <Q 𝑢 ↔ (𝐹‘𝑞) <Q 𝑢))
18	17	abbidv 2217	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝐹‘𝑥) <Q 𝑢} = {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢})
19	16, 18	opeq12d 3660	. . . . 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 5713	. . . . . . . . 9 ⊢ (𝑥 = 𝑞 → (𝑥 +Q 𝑦) = (𝑞 +Q 𝑦))
21	20	breq2d 3887	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝑙 <Q (𝑥 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑦)))
22	21	abbidv 2217	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q (𝑥 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)})
23	20	breq1d 3885	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝑥 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑦) <Q 𝑢))
24	23	abbidv 2217	. . . . . . 7 ⊢ (𝑥 = 𝑞 → {𝑢 ∣ (𝑥 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢})
25	22, 24	opeq12d 3660	. . . . . 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 5722	. . . . 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 3888	. . . 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 5724	. . . . . . . 8 ⊢ (𝑥 = 𝑞 → ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)))
29	28	breq2d 3887	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))))
30	29	abbidv 2217	. . . . . 6 ⊢ (𝑥 = 𝑞 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑥) +Q (𝑥 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))})
31	28	breq1d 3885	. . . . . . 7 ⊢ (𝑥 = 𝑞 → (((𝐹‘𝑥) +Q (𝑥 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢))
32	31	abbidv 2217	. . . . . 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 3660	. . . . 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 3887	. . . 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 460	. . 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 5714	. . . . . . . . 9 ⊢ (𝑦 = 𝑟 → (𝑞 +Q 𝑦) = (𝑞 +Q 𝑟))
37	36	breq2d 3887	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑙 <Q (𝑞 +Q 𝑦) ↔ 𝑙 <Q (𝑞 +Q 𝑟)))
38	37	abbidv 2217	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑦)} = {𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)})
39	36	breq1d 3885	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝑞 +Q 𝑦) <Q 𝑢 ↔ (𝑞 +Q 𝑟) <Q 𝑢))
40	39	abbidv 2217	. . . . . . 7 ⊢ (𝑦 = 𝑟 → {𝑢 ∣ (𝑞 +Q 𝑦) <Q 𝑢} = {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢})
41	38, 40	opeq12d 3660	. . . . . 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 5722	. . . . 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 3887	. . . 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 5722	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) = ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)))
45	44	breq2d 3887	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) ↔ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))))
46	45	abbidv 2217	. . . . . 6 ⊢ (𝑦 = 𝑟 → {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑦))} = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))})
47	44	breq1d 3885	. . . . . . 7 ⊢ (𝑦 = 𝑟 → (((𝐹‘𝑞) +Q (𝑞 +Q 𝑦)) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢))
48	47	abbidv 2217	. . . . . 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 3660	. . . . 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 3887	. . . 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 460	. . 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 2620	. 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 1299   ∈ wcel 1448  {cab 2086  ∀wral 2375  ∃wrex 2376  {crab 2379  ⟨cop 3477   class class class wbr 3875  ⟶wf 5055  ‘cfv 5059  (class class class)co 5706  Qcnq 6989   +_Q cplq 6991   <_Q cltq 6994   +_P cpp 7002  <_P cltp 7004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-iltp 7179

This theorem is referenced by:  cauappcvgpr  7371