Theorem cauappcvgpr 7865

Description: A Cauchy approximation has a limit. A Cauchy approximation, here 𝐹, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of 𝐹 is Q rather than P. We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 7885 and caucvgprpr 7915 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-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))

Assertion

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

Distinct variable groups:   𝐴,𝑝   𝐹,𝑞,𝑦,𝑟,𝑢   𝐹,𝑝,𝑙,𝑞   𝑦,𝑙,𝑟   𝑢,𝑞,𝑦,𝑟   𝑢,𝑝,𝑟,𝑞,𝑙   𝜑,𝑞,𝑝

Allowed substitution hints:   𝜑(𝑦,𝑢,𝑟,𝑙)   𝐴(𝑦,𝑢,𝑟,𝑞,𝑙)

Proof of Theorem cauappcvgpr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . 3 ⊢ (𝜑 → 𝐹:Q⟶Q)
2		cauappcvgpr.app	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
3		cauappcvgpr.bnd	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		oveq2 6018	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → (𝑙 +Q 𝑧) = (𝑙 +Q 𝑞))
5		fveq2 5632	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → (𝐹‘𝑧) = (𝐹‘𝑞))
6	4, 5	breq12d 4096	. . . . . . 7 ⊢ (𝑧 = 𝑞 → ((𝑙 +Q 𝑧) <Q (𝐹‘𝑧) ↔ (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)))
7	6	cbvrexv 2766	. . . . . 6 ⊢ (∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧) ↔ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞))
8	7	a1i 9	. . . . 5 ⊢ (𝑙 ∈ Q → (∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧) ↔ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)))
9	8	rabbiia 2784	. . . 4 ⊢ {𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)} = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
10		id 19	. . . . . . . . 9 ⊢ (𝑧 = 𝑞 → 𝑧 = 𝑞)
11	5, 10	oveq12d 6028	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → ((𝐹‘𝑧) +Q 𝑧) = ((𝐹‘𝑞) +Q 𝑞))
12	11	breq1d 4093	. . . . . . 7 ⊢ (𝑧 = 𝑞 → (((𝐹‘𝑧) +Q 𝑧) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢))
13	12	cbvrexv 2766	. . . . . 6 ⊢ (∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢)
14	13	a1i 9	. . . . 5 ⊢ (𝑢 ∈ Q → (∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢))
15	14	rabbiia 2784	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
16	9, 15	opeq12i 3862	. . 3 ⊢ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
17	1, 2, 3, 16	cauappcvgprlemcl 7856	. 2 ⊢ (𝜑 → ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ ∈ P)
18	1, 2, 3, 16	cauappcvgprlemlim 7864	. 2 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
19		oveq1 6017	. . . . . 6 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ → (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) = (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩))
20	19	breq2d 4095	. . . . 5 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩)))
21		breq1 4086	. . . . 5 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ → (𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩ ↔ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
22	20, 21	anbi12d 473	. . . 4 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ → ((⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)))
23	22	2ralbidv 2554	. . 3 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ → (∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩) ↔ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)))
24	23	rspcev 2907	. 2 ⊢ ((⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ ∈ P ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ Q (𝑙 +Q 𝑧) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ Q ((𝐹‘𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)) → ∃𝑦 ∈ P ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
25	17, 18, 24	syl2anc 411	1 ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<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   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  {crab 2512  ⟨cop 3669   class class class wbr 4083  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010  Qcnq 7483   +_Q cplq 7485   <_Q cltq 7488  Pcnp 7494   +_P cpp 7496  <_P cltp 7498

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4381  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-1o 6573  df-2o 6574  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-pli 7508  df-mi 7509  df-lti 7510  df-plpq 7547  df-mpq 7548  df-enq 7550  df-nqqs 7551  df-plqqs 7552  df-mqqs 7553  df-1nqqs 7554  df-rq 7555  df-ltnqqs 7556  df-enq0 7627  df-nq0 7628  df-0nq0 7629  df-plq0 7630  df-mq0 7631  df-inp 7669  df-iplp 7671  df-iltp 7673

This theorem is referenced by: (None)