Theorem caucvgpr 7490

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 7470 and caucvgprpr 7520. Reading cauappcvgpr 7470 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 3705	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑗 → ⟨𝑧, 1o⟩ = ⟨𝑗, 1o⟩)
5	4	eceq1d 6465	. . . . . . . . . 10 ⊢ (𝑧 = 𝑗 → [⟨𝑧, 1o⟩] ~Q = [⟨𝑗, 1o⟩] ~Q )
6	5	fveq2d 5425	. . . . . . . . 9 ⊢ (𝑧 = 𝑗 → (*Q‘[⟨𝑧, 1o⟩] ~Q ) = (*Q‘[⟨𝑗, 1o⟩] ~Q ))
7	6	oveq2d 5790	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
8		fveq2 5421	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝐹‘𝑧) = (𝐹‘𝑗))
9	7, 8	breq12d 3942	. . . . . . 7 ⊢ (𝑧 = 𝑗 → ((𝑙 +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q (𝐹‘𝑧) ↔ (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
10	9	cbvrexv 2655	. . . . . 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 2671	. . . 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 5792	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → ((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) = ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
14	13	breq1d 3939	. . . . . . 7 ⊢ (𝑧 = 𝑗 → (((𝐹‘𝑧) +Q (*Q‘[⟨𝑧, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢))
15	14	cbvrexv 2655	. . . . . 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 2671	. . . 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 3710	. . 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 7484	. 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 7489	. 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 5781	. . . . . . . 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 3941	. . . . . . 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 3932	. . . . . . 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 464	. . . . . 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 2461	. . . 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 2437	. . 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 2789	. 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 408	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 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  [cec 6427  Ncnpi 7080   <_N clti 7083   ~_Q ceq 7087  Qcnq 7088   +_Q cplq 7090  *_Qcrq 7092   <_Q cltq 7093  Pcnp 7099   +_P cpp 7101  <_P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by: (None)