Theorem cauappcvgpr 7675

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 7695 and caucvgprpr 7725 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 5896	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → (𝑙 +Q 𝑧) = (𝑙 +Q 𝑞))
5		fveq2 5527	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → (𝐹‘𝑧) = (𝐹‘𝑞))
6	4, 5	breq12d 4028	. . . . . . 7 ⊢ (𝑧 = 𝑞 → ((𝑙 +Q 𝑧) <Q (𝐹‘𝑧) ↔ (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)))
7	6	cbvrexv 2716	. . . . . 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 2734	. . . 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 5906	. . . . . . . 8 ⊢ (𝑧 = 𝑞 → ((𝐹‘𝑧) +Q 𝑧) = ((𝐹‘𝑞) +Q 𝑞))
12	11	breq1d 4025	. . . . . . 7 ⊢ (𝑧 = 𝑞 → (((𝐹‘𝑧) +Q 𝑧) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢))
13	12	cbvrexv 2716	. . . . . 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 2734	. . . 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 3795	. . 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 7666	. 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 7674	. 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 5895	. . . . . 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 4027	. . . . 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 4018	. . . . 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 2511	. . 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 2853	. 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 1363   ∈ wcel 2158  {cab 2173  ∀wral 2465  ∃wrex 2466  {crab 2469  ⟨cop 3607   class class class wbr 4015  ⟶wf 5224  ‘cfv 5228  (class class class)co 5888  Qcnq 7293   +_Q cplq 7295   <_Q cltq 7298  Pcnp 7304   +_P cpp 7306  <_P cltp 7308

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479  df-iplp 7481  df-iltp 7483

This theorem is referenced by: (None)