Theorem caucvgpr 7302

Description: A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). 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 cauappcvgpr 7282 and caucvgprpr 7332. Reading cauappcvgpr 7282 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))

Assertion

Ref	Expression
caucvgpr	⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑘,𝑛,𝑙,𝑢,𝑥,𝑦   𝜑,𝑗,𝑘,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑢,𝑛,𝑙)   𝐴(𝑥,𝑦,𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgpr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . 3 ⊢ (𝜑 → 𝐹:N⟶Q)
2		caucvgpr.cau	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
3		caucvgpr.bnd	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
4		opeq1 3628	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑗 → ⟨𝑧, 1o⟩ = ⟨𝑗, 1o⟩)
5	4	eceq1d 6342	. . . . . . . . . 10 ⊢ (𝑧 = 𝑗 → [⟨𝑧, 1o⟩] ~Q = [⟨𝑗, 1o⟩] ~Q )
6	5	fveq2d 5322	. . . . . . . . 9 ⊢ (𝑧 = 𝑗 → (*Q‘[⟨𝑧, 1o⟩] ~Q ) = (*Q‘[⟨𝑗, 1o⟩] ~Q ))
7	6	oveq2d 5682	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
8		fveq2 5318	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝐹‘𝑧) = (𝐹‘𝑗))
9	7, 8	breq12d 3864	. . . . . . 7 ⊢ (𝑧 = 𝑗 → ((𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧) ↔ (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
10	9	cbvrexv 2592	. . . . . 6 ⊢ (∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧) ↔ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
11	10	a1i 9	. . . . 5 ⊢ (𝑙 ∈ Q → (∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧) ↔ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
12	11	rabbiia 2605	. . . 4 ⊢ {𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)} = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
13	8, 6	oveq12d 5684	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) = ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
14	13	breq1d 3861	. . . . . . 7 ⊢ (𝑧 = 𝑗 → (((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢))
15	14	cbvrexv 2592	. . . . . 6 ⊢ (∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢)
16	15	a1i 9	. . . . 5 ⊢ (𝑢 ∈ Q → (∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢))
17	16	rabbiia 2605	. . . 4 ⊢ {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
18	12, 17	opeq12i 3633	. . 3 ⊢ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
19	1, 2, 3, 18	caucvgprlemcl 7296	. 2 ⊢ (𝜑 → ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
20	1, 2, 3, 18	caucvgprlemlim 7301	. 2 ⊢ (𝜑 → ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
21		oveq1 5673	. . . . . . . 8 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) = (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩))
22	21	breq2d 3863	. . . . . . 7 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩)))
23		breq1 3854	. . . . . . 7 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → (𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩ ↔ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))
24	22, 23	anbi12d 458	. . . . . 6 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → ((⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
25	24	imbi2d 229	. . . . 5 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → ((𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)) ↔ (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))))
26	25	rexralbidv 2405	. . . 4 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → (∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)) ↔ ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))))
27	26	ralbidv 2381	. . 3 ⊢ (𝑦 = ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ → (∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)) ↔ ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))))
28	27	rspcev 2723	. 2 ⊢ ((⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ ∈ P ∧ ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙 ∈ Q ∣ ∃𝑧 ∈ N (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧)}, {𝑢 ∈ Q ∣ ∃𝑧 ∈ N ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩))) → ∃𝑦 ∈ P ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
29	19, 20, 28	syl2anc 404	1 ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290   ∈ wcel 1439  {cab 2075  ∀wral 2360  ∃wrex 2361  {crab 2364  ⟨cop 3453   class class class wbr 3851  ⟶wf 5024  ‘cfv 5028  (class class class)co 5666  1_oc1o 6188  [cec 6304  Ncnpi 6892   <_N clti 6895   ~_Q ceq 6899  Qcnq 6900   +_Q cplq 6902  *_Qcrq 6904   <_Q cltq 6905  Pcnp 6911   +_P cpp 6913  <_P cltp 6915

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-iplp 7088  df-iltp 7090

This theorem is referenced by: (None)