Theorem cauappcvgpr 7837

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 7857 and caucvgprpr 7887 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 6002	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → (𝑙 +Q 𝑧) = (𝑙 +Q 𝑞))
5		fveq2 5623	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → (𝐹‘𝑧) = (𝐹‘𝑞))
6	4, 5	breq12d 4095	. . . . . . 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 6012	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → ((𝐹‘𝑧) +Q 𝑧) = ((𝐹‘𝑞) +Q 𝑞))
12	11	breq1d 4092	. . . . . . 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 3861	. . 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 7828	. 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 7836	. 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 6001	. . . . . 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 4094	. . . . 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 4085	. . . . 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 4082  ⟶wf 5310  ‘cfv 5314  (class class class)co 5994  Qcnq 7455   +_Q cplq 7457   <_Q cltq 7460  Pcnp 7466   +_P cpp 7468  <_P cltp 7470

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-2o 6553  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-enq0 7599  df-nq0 7600  df-0nq0 7601  df-plq0 7602  df-mq0 7603  df-inp 7641  df-iplp 7643  df-iltp 7645

This theorem is referenced by: (None)