Theorem caucvgsr 7764

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 7674 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 7763).

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 7759).

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 7674 to get a limit (see caucvgsrlemgt1 7757).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7757).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7762). (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 3992	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘))
4		fveq2 5496	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o))
5		opeq1 3765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → ⟨𝑛, 1o⟩ = ⟨1o, 1o⟩)
6	5	eceq1d 6549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [⟨𝑛, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
7	6	fveq2d 5500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
8	7	breq2d 4001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
9	8	abbidv 2288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
10	7	breq1d 3999	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢))
11	10	abbidv 2288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢})
12	9, 11	opeq12d 3773	. . . . . . . . . . . . . . . . . . 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 5868	. . . . . . . . . . . . . . . . . 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 3771	. . . . . . . . . . . . . . . . 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 6549	. . . . . . . . . . . . . . . 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 5869	. . . . . . . . . . . . . . 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 4002	. . . . . . . . . . . . . 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 5871	. . . . . . . . . . . . . . 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 4001	. . . . . . . . . . . . . 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 470	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 2470	. . . . . . . . . . 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 7277	. . . . . . . . . . . 12 ⊢ 1o ∈ N
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N)
25	22, 2, 24	rspcdva 2839	. . . . . . . . . 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 108	. . . . . . . . . . . 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 2533	. . . . . . . . . 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 3993	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚))
31		fveq2 5496	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
32	31	oveq1d 5868	. . . . . . . . . . . 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 4001	. . . . . . . . . . 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 233	. . . . . . . . . 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 2830	. . . . . . . . 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 279	. . . . . . . 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 7313	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
38	37	fveq2i 5499	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
39		rec1nq 7357	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q
40	38, 39	eqtr3i 2193	. . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
41	40	breq2i 3997	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑙 <Q 1Q)
42	41	abbii 2286	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q}
43	40	breq1i 3996	. . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
44	43	abbii 2286	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
45	42, 44	opeq12i 3770	. . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . 15 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
47	45, 46	eqtr4i 2194	. . . . . . . . . . . . . 14 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = 1P
48	47	oveq1i 5863	. . . . . . . . . . . . 13 ⊢ (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
49	48	opeq1i 3768	. . . . . . . . . . . 12 ⊢ ⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩
50		eceq1 6548	. . . . . . . . . . . 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 7694	. . . . . . . . . . 11 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
53	51, 52	eqtr4i 2194	. . . . . . . . . 10 ⊢ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
54	53	oveq2i 5864	. . . . . . . . 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 3997	. . . . . . . 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	syl6ib 160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)))
57	56	imp 123	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o <N 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
58	1	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 𝐹:N⟶R)
59	23	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → 1o ∈ N)
60	58, 59	ffvelrnd 5632	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R)
61		ltadd1sr 7738	. . . . . . . . 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 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
64		fveq2 5496	. . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚))
65	64	oveq1d 5868	. . . . . . . 8 ⊢ (1o = 𝑚 → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R))
66	65	adantl 275	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → ((𝐹‘1o) +R 1R) = ((𝐹‘𝑚) +R 1R))
67	63, 66	breqtrd 4015	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
68		nlt1pig 7303	. . . . . . . . 9 ⊢ (𝑚 ∈ N → ¬ 𝑚 <N 1o)
69	68	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ¬ 𝑚 <N 1o)
70	69	pm2.21d 614	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝑚 <N 1o → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R)))
71	70	imp 123	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 𝑚 <N 1o) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
72		pitri3or 7284	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o))
73	23, 72	mpan 422	. . . . . . 7 ⊢ (𝑚 ∈ N → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o))
74	73	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (1o <N 𝑚 ∨ 1o = 𝑚 ∨ 𝑚 <N 1o))
75	57, 67, 71, 74	mpjao3dan 1302	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
76		ltasrg 7732	. . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
77	76	adantl 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
78	1	ffvelrnda 5631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R)
79		1sr 7713	. . . . . . 7 ⊢ 1R ∈ R
80		addclsr 7715	. . . . . . 7 ⊢ (((𝐹‘𝑚) ∈ R ∧ 1R ∈ R) → ((𝐹‘𝑚) +R 1R) ∈ R)
81	78, 79, 80	sylancl 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R 1R) ∈ R)
82		m1r 7714	. . . . . . 7 ⊢ -1R ∈ R
83	82	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R)
84		addcomsrg 7717	. . . . . . 7 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
85	84	adantl 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
86	77, 60, 81, 83, 85	caovord2d 6022	. . . . 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 146	. . . 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 7718	. . . . . 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 1233	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R)))
91		addcomsrg 7717	. . . . . . . . 9 ⊢ ((1R ∈ R ∧ -1R ∈ R) → (1R +R -1R) = (-1R +R 1R))
92	79, 82, 91	mp2an 424	. . . . . . . 8 ⊢ (1R +R -1R) = (-1R +R 1R)
93		m1p1sr 7722	. . . . . . . 8 ⊢ (-1R +R 1R) = 0R
94	92, 93	eqtri 2191	. . . . . . 7 ⊢ (1R +R -1R) = 0R
95	94	oveq2i 5864	. . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R)
96		0idsr 7729	. . . . . . 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 2215	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚))
99	90, 98	eqtrd 2203	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚))
100	87, 99	breqtrd 4015	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
101	100	ralrimiva 2543	. 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
102	1, 2, 101	caucvgsrlembnd 7763	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 103   ↔ wb 104   ∨ w3o 972   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  ⟨cop 3586   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  [cec 6511  Ncnpi 7234   <_N clti 7237   ~_Q ceq 7241  1_Qc1q 7243  *_Qcrq 7246   <_Q cltq 7247  1_Pc1p 7254   +_P cpp 7255   ~_R cer 7258  Rcnr 7259  0_Rc0r 7260  1_Rc1r 7261  -1_Rcm1r 7262   +_R cplr 7263   <_R cltr 7265

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-imp 7431  df-iltp 7432  df-enr 7688  df-nr 7689  df-plr 7690  df-mr 7691  df-ltr 7692  df-0r 7693  df-1r 7694  df-m1r 7695

This theorem is referenced by:  axcaucvglemres  7861