Theorem caucvgsr 8116

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 8026 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 8115).

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 8111).

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 8026 to get a limit (see caucvgsrlemgt1 8109).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8109).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8114). (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 4111	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘))
4		fveq2 5669	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o))
5		opeq1 3882	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → ⟨𝑛, 1o⟩ = ⟨1o, 1o⟩)
6	5	eceq1d 6802	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [⟨𝑛, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
7	6	fveq2d 5673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
8	7	breq2d 4120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
9	8	abbidv 2352	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
10	7	breq1d 4118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢))
11	10	abbidv 2352	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢})
12	9, 11	opeq12d 3890	. . . . . . . . . . . . . . . . . . 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 6064	. . . . . . . . . . . . . . . . . 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 3888	. . . . . . . . . . . . . . . . 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 6802	. . . . . . . . . . . . . . . 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 6065	. . . . . . . . . . . . . . 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 4121	. . . . . . . . . . . . . 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 6067	. . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . 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 2542	. . . . . . . . . . 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 7629	. . . . . . . . . . . 12 ⊢ 1o ∈ N
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N)
25	22, 2, 24	rspcdva 2925	. . . . . . . . . 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 2605	. . . . . . . . . 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 4112	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚))
31		fveq2 5669	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
32	31	oveq1d 6064	. . . . . . . . . . . 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 4120	. . . . . . . . . . 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 2916	. . . . . . . . 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 7665	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
38	37	fveq2i 5672	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
39		rec1nq 7709	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q
40	38, 39	eqtr3i 2255	. . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
41	40	breq2i 4116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑙 <Q 1Q)
42	41	abbii 2348	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q}
43	40	breq1i 4115	. . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
44	43	abbii 2348	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
45	42, 44	opeq12i 3887	. . . . . . . . . . . . . . 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 7781	. . . . . . . . . . . . . . 15 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
47	45, 46	eqtr4i 2256	. . . . . . . . . . . . . 14 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = 1P
48	47	oveq1i 6059	. . . . . . . . . . . . 13 ⊢ (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
49	48	opeq1i 3885	. . . . . . . . . . . 12 ⊢ ⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩
50		eceq1 6801	. . . . . . . . . . . 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 8046	. . . . . . . . . . 11 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
53	51, 52	eqtr4i 2256	. . . . . . . . . 10 ⊢ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
54	53	oveq2i 6060	. . . . . . . . 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 4116	. . . . . . . 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 5812	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R)
61		ltadd1sr 8090	. . . . . . . . 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 5669	. . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚))
65	64	oveq1d 6064	. . . . . . . 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 4134	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
68		nlt1pig 7655	. . . . . . . . 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 7636	. . . . . . . 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 8084	. . . . . . 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 5811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R)
79		1sr 8065	. . . . . . 7 ⊢ 1R ∈ R
80		addclsr 8067	. . . . . . 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 8066	. . . . . . 7 ⊢ -1R ∈ R
83	82	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R)
84		addcomsrg 8069	. . . . . . 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 6223	. . . . 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 8070	. . . . . 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 8069	. . . . . . . . 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 8074	. . . . . . . 8 ⊢ (-1R +R 1R) = 0R
94	92, 93	eqtri 2253	. . . . . . 7 ⊢ (1R +R -1R) = 0R
95	94	oveq2i 6060	. . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R)
96		0idsr 8081	. . . . . . 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 2277	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚))
99	90, 98	eqtrd 2265	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚))
100	87, 99	breqtrd 4134	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
101	100	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
102	1, 2, 101	caucvgsrlembnd 8115	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 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  ⟨cop 3691   class class class wbr 4108  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  1_oc1o 6639  [cec 6764  Ncnpi 7586   <_N clti 7589   ~_Q ceq 7593  1_Qc1q 7595  *_Qcrq 7598   <_Q cltq 7599  1_Pc1p 7606   +_P cpp 7607   ~_R cer 7610  Rcnr 7611  0_Rc0r 7612  1_Rc1r 7613  -1_Rcm1r 7614   +_R cplr 7615   <_R cltr 7617

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-enq0 7738  df-nq0 7739  df-0nq0 7740  df-plq0 7741  df-mq0 7742  df-inp 7780  df-i1p 7781  df-iplp 7782  df-imp 7783  df-iltp 7784  df-enr 8040  df-nr 8041  df-plr 8042  df-mr 8043  df-ltr 8044  df-0r 8045  df-1r 8046  df-m1r 8047

This theorem is referenced by:  axcaucvglemres  8213