Theorem caucvgsr 8134

Description: A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). 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).

This is similar to caucvgprpr 8044 but is for signed reals rather than positive reals.

Here is an outline of how we prove it:

1. Choose a lower bound for the sequence (see caucvgsrlembnd 8133).

2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 8129).

3. Since a signed real (element of R) which is greater than zero can be mapped to a positive real (element of P), perform that mapping on each element of the sequence and invoke caucvgprpr 8044 to get a limit (see caucvgsrlemgt1 8127).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8127).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8132). (Contributed by Jim Kingdon, 20-Jun-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	⊢ (𝜑 → 𝐹:N⟶R)
caucvgsr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))

Assertion

Ref	Expression
caucvgsr	⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))

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

Allowed substitution hints:   𝜑(𝑦,𝑢,𝑙)

Proof of Theorem caucvgsr

Dummy variables 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgsr.f	. 2 ⊢ (𝜑 → 𝐹:N⟶R)
2		caucvgsr.cau	. 2 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
3		breq1 4118	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘))
4		fveq2 5676	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o))
5		opeq1 3889	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → ⟨𝑛, 1o⟩ = ⟨1o, 1o⟩)
6	5	eceq1d 6817	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [⟨𝑛, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
7	6	fveq2d 5680	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
8	7	breq2d 4127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
9	8	abbidv 2354	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
10	7	breq1d 4125	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢))
11	10	abbidv 2354	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢})
12	9, 11	opeq12d 3897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1o → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩)
13	12	oveq1d 6074	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1o → (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P))
14	13	opeq1d 3895	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1o → ⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩)
15	14	eceq1d 6817	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1o → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
16	15	oveq2d 6075	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
17	4, 16	breq12d 4128	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1o → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
18	4, 15	oveq12d 6077	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
19	18	breq2d 4127	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1o → ((𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
20	17, 19	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ↔ ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
21	3, 20	imbi12d 234	. . . . . . . . . . . 12 ⊢ (𝑛 = 1o → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) ↔ (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
22	21	ralbidv 2544	. . . . . . . . . . 11 ⊢ (𝑛 = 1o → (∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) ↔ ∀𝑘 ∈ N (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
23		1pi 7647	. . . . . . . . . . . 12 ⊢ 1o ∈ N
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N)
25	22, 2, 24	rspcdva 2928	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ N (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
26		simpl 109	. . . . . . . . . . . 12 ⊢ (((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
27	26	imim2i 12	. . . . . . . . . . 11 ⊢ ((1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
28	27	ralimi 2607	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ N (1o <N 𝑘 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘1o) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → ∀𝑘 ∈ N (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
29	25, 28	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ N (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
30		breq2 4119	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚))
31		fveq2 5676	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
32	31	oveq1d 6074	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
33	32	breq2d 4127	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘1o) <R ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
34	30, 33	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ↔ (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
35	34	rspcv 2919	. . . . . . . . 9 ⊢ (𝑚 ∈ N → (∀𝑘 ∈ N (1o <N 𝑘 → (𝐹‘1o) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
36	29, 35	mpan9 281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
37		df-1nqqs 7683	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
38	37	fveq2i 5679	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
39		rec1nq 7727	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q
40	38, 39	eqtr3i 2257	. . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
41	40	breq2i 4123	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑙 <Q 1Q)
42	41	abbii 2350	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q}
43	40	breq1i 4122	. . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
44	43	abbii 2350	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
45	42, 44	opeq12i 3894	. . . . . . . . . . . . . . 15 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46		df-i1p 7799	. . . . . . . . . . . . . . 15 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
47	45, 46	eqtr4i 2258	. . . . . . . . . . . . . 14 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = 1P
48	47	oveq1i 6069	. . . . . . . . . . . . 13 ⊢ (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
49	48	opeq1i 3892	. . . . . . . . . . . 12 ⊢ ⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩
50		eceq1 6816	. . . . . . . . . . . 12 ⊢ (⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩ → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 ⊢ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R
52		df-1r 8064	. . . . . . . . . . 11 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
53	51, 52	eqtr4i 2258	. . . . . . . . . 10 ⊢ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
54	53	oveq2i 6070	. . . . . . . . 9 ⊢ ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹‘𝑚) +R 1R)
55	54	breq2i 4123	. . . . . . . 8 ⊢ ((𝐹‘1o) <R ((𝐹‘𝑚) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
56	36, 55	imbitrdi 161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)))
57	56	imp 124	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o <N 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
58	1	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐹:N⟶R)
59	23	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1o ∈ N)
60	58, 59	ffvelcdmd 5819	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R)
61		ltadd1sr 8108	. . . . . . . . 9 ⊢ ((𝐹‘1o) ∈ R → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
62	60, 61	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
63	62	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
64		fveq2 5676	. . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚))
65	64	oveq1d 6074	. . . . . . . 8 ⊢ (1o = 𝑚 → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R))
66	65	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R))
67	63, 66	breqtrd 4141	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
68		nlt1pig 7673	. . . . . . . . 9 ⊢ (𝑚 ∈ N → ¬ 𝑚 <N 1o)
69	68	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ¬ 𝑚 <N 1o)
70	69	pm2.21d 624	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝑚 <N 1o → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)))
71	70	imp 124	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 𝑚 <N 1o) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
72		pitri3or 7654	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o))
73	23, 72	mpan 424	. . . . . . 7 ⊢ (𝑚 ∈ N → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o))
74	73	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o))
75	57, 67, 71, 74	mpjao3dan 1344	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
76		ltasrg 8102	. . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
77	76	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
78	1	ffvelcdmda 5818	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R)
79		1sr 8083	. . . . . . 7 ⊢ 1R ∈ R
80		addclsr 8085	. . . . . . 7 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑚) +R 1R) ∈ R)
81	78, 79, 80	sylancl 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 1R) ∈ R)
82		m1r 8084	. . . . . . 7 ⊢ -1R ∈ R
83	82	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R)
84		addcomsrg 8087	. . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
85	84	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
86	77, 60, 81, 83, 85	caovord2d 6233	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) <R ((𝐹‘𝑚) +R 1R) ↔ ((𝐹‘1o) +R -1R) <R (((𝐹‘𝑚) +R 1R) +R -1R)))
87	75, 86	mpbid 147	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (((𝐹‘𝑚) +R 1R) +R -1R))
88	79	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1R ∈ R)
89		addasssrg 8088	. . . . . 6 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R ∧ -1R ∈ R) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R)))
90	78, 88, 83, 89	syl3anc 1274	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R)))
91		addcomsrg 8087	. . . . . . . . 9 ⊢ ((1R ∈ R ∧ -1R ∈ R) → (1R +R -1R) = (-1R +R 1R))
92	79, 82, 91	mp2an 426	. . . . . . . 8 ⊢ (1R +R -1R) = (-1R +R 1R)
93		m1p1sr 8092	. . . . . . . 8 ⊢ (-1R +R 1R) = 0R
94	92, 93	eqtri 2255	. . . . . . 7 ⊢ (1R +R -1R) = 0R
95	94	oveq2i 6070	. . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R)
96		0idsr 8099	. . . . . . 7 ⊢ ((𝐹‘𝑚) ∈ R → ((𝐹‘𝑚) +R 0R) = (𝐹‘𝑚))
97	78, 96	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 0R) = (𝐹‘𝑚))
98	95, 97	eqtrid 2279	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚))
99	90, 98	eqtrd 2267	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚))
100	87, 99	breqtrd 4141	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
101	100	ralrimiva 2617	. 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
102	1, 2, 101	caucvgsrlembnd 8133	1 ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1004   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  ⟨cop 3698   class class class wbr 4115  ⟶wf 5354  ‘cfv 5358  (class class class)co 6059  1_oc1o 6654  [cec 6779  Ncnpi 7604   <_N clti 7607   ~_Q ceq 7611  1_Qc1q 7613  *_Qcrq 7616   <_Q cltq 7617  1_Pc1p 7624   +_P cpp 7625   ~_R cer 7628  Rcnr 7629  0_Rc0r 7630  1_Rc1r 7631  -1_Rcm1r 7632   +_R cplr 7633   <_R cltr 7635

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-eprel 4416  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-irdg 6615  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6781  df-ec 6783  df-qs 6787  df-ni 7636  df-pli 7637  df-mi 7638  df-lti 7639  df-plpq 7676  df-mpq 7677  df-enq 7679  df-nqqs 7680  df-plqqs 7681  df-mqqs 7682  df-1nqqs 7683  df-rq 7684  df-ltnqqs 7685  df-enq0 7756  df-nq0 7757  df-0nq0 7758  df-plq0 7759  df-mq0 7760  df-inp 7798  df-i1p 7799  df-iplp 7800  df-imp 7801  df-iltp 7802  df-enr 8058  df-nr 8059  df-plr 8060  df-mr 8061  df-ltr 8062  df-0r 8063  df-1r 8064  df-m1r 8065

This theorem is referenced by:  axcaucvglemres  8231