Theorem caucvgpr 7656

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 7636 and caucvgprpr 7686. Reading cauappcvgpr 7636 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 3774	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑗 → ⟨𝑧, 1o⟩ = ⟨𝑗, 1o⟩)
5	4	eceq1d 6561	. . . . . . . . . 10 ⊢ (𝑧 = 𝑗 → [⟨𝑧, 1o⟩] ~Q = [⟨𝑗, 1o⟩] ~Q )
6	5	fveq2d 5511	. . . . . . . . 9 ⊢ (𝑧 = 𝑗 → (*Q‘[⟨𝑧, 1o⟩] ~Q ) = (*Q‘[⟨𝑗, 1o⟩] ~Q ))
7	6	oveq2d 5881	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
8		fveq2 5507	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝐹‘𝑧) = (𝐹‘𝑗))
9	7, 8	breq12d 4011	. . . . . . 7 ⊢ (𝑧 = 𝑗 → ((𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧) ↔ (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
10	9	cbvrexv 2702	. . . . . 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 2720	. . . 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 5883	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) = ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
14	13	breq1d 4008	. . . . . . 7 ⊢ (𝑧 = 𝑗 → (((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢))
15	14	cbvrexv 2702	. . . . . 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 2720	. . . 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 3779	. . 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 7650	. 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 7655	. 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 5872	. . . . . . . 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 4010	. . . . . . 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 4001	. . . . . . 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 473	. . . . . 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 230	. . . . 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 2501	. . . 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 2475	. . 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 2839	. 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 411	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 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  {cab 2161  ∀wral 2453  ∃wrex 2454  {crab 2457  ⟨cop 3592   class class class wbr 3998  ⟶wf 5204  ‘cfv 5208  (class class class)co 5865  1_oc1o 6400  [cec 6523  Ncnpi 7246   <_N clti 7249   ~_Q ceq 7253  Qcnq 7254   +_Q cplq 7256  *_Qcrq 7258   <_Q cltq 7259  Pcnp 7265   +_P cpp 7267  <_P cltp 7269

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-iplp 7442  df-iltp 7444

This theorem is referenced by: (None)