Theorem caucvgsr 8022

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 7932 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 8021).

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 8017).

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 7932 to get a limit (see caucvgsrlemgt1 8015).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 8015).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 8020). (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 4091	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘))
4		fveq2 5639	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1o → (𝐹‘𝑛) = (𝐹‘1o))
5		opeq1 3862	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 1o → ⟨𝑛, 1o⟩ = ⟨1o, 1o⟩)
6	5	eceq1d 6738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 1o → [⟨𝑛, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
7	6	fveq2d 5643	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 1o → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
8	7	breq2d 4100	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
9	8	abbidv 2349	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
10	7	breq1d 4098	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1o → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢))
11	10	abbidv 2349	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1o → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢})
12	9, 11	opeq12d 3870	. . . . . . . . . . . . . . . . . . 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 6033	. . . . . . . . . . . . . . . . . 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 3868	. . . . . . . . . . . . . . . . 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 6738	. . . . . . . . . . . . . . . 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 6034	. . . . . . . . . . . . . . 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 4101	. . . . . . . . . . . . . 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 6036	. . . . . . . . . . . . . . 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 4100	. . . . . . . . . . . . . 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 2532	. . . . . . . . . . 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 7535	. . . . . . . . . . . 12 ⊢ 1o ∈ N
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → 1o ∈ N)
25	22, 2, 24	rspcdva 2915	. . . . . . . . . 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 2595	. . . . . . . . . 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 4092	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚))
31		fveq2 5639	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
32	31	oveq1d 6033	. . . . . . . . . . . 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 4100	. . . . . . . . . . 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 2906	. . . . . . . . 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 7571	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
38	37	fveq2i 5642	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
39		rec1nq 7615	. . . . . . . . . . . . . . . . . . 19 ⊢ (*Q‘1Q) = 1Q
40	38, 39	eqtr3i 2254	. . . . . . . . . . . . . . . . . 18 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
41	40	breq2i 4096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑙 <Q 1Q)
42	41	abbii 2347	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q 1Q}
43	40	breq1i 4095	. . . . . . . . . . . . . . . . 17 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
44	43	abbii 2347	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
45	42, 44	opeq12i 3867	. . . . . . . . . . . . . . 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 7687	. . . . . . . . . . . . . . 15 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
47	45, 46	eqtr4i 2255	. . . . . . . . . . . . . 14 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = 1P
48	47	oveq1i 6028	. . . . . . . . . . . . 13 ⊢ (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
49	48	opeq1i 3865	. . . . . . . . . . . 12 ⊢ ⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩
50		eceq1 6737	. . . . . . . . . . . 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 7952	. . . . . . . . . . 11 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
53	51, 52	eqtr4i 2255	. . . . . . . . . 10 ⊢ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
54	53	oveq2i 6029	. . . . . . . . 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 4096	. . . . . . . 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 5783	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) ∈ R)
61		ltadd1sr 7996	. . . . . . . . 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 5639	. . . . . . . . 9 ⊢ (1o = 𝑚 → (𝐹‘1o) = (𝐹‘𝑚))
65	64	oveq1d 6033	. . . . . . . 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 4114	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
68		nlt1pig 7561	. . . . . . . . 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 7542	. . . . . . . 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 1343	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘1o) <R ((𝐹‘𝑚) +R 1R))
76		ltasrg 7990	. . . . . . 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 5782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (𝐹‘𝑚) ∈ R)
79		1sr 7971	. . . . . . 7 ⊢ 1R ∈ R
80		addclsr 7973	. . . . . . 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 7972	. . . . . . 7 ⊢ -1R ∈ R
83	82	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → -1R ∈ R)
84		addcomsrg 7975	. . . . . . 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 6192	. . . . 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 7976	. . . . . 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 1273	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = ((𝐹‘𝑚) +R (1R +R -1R)))
91		addcomsrg 7975	. . . . . . . . 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 7980	. . . . . . . 8 ⊢ (-1R +R 1R) = 0R
94	92, 93	eqtri 2252	. . . . . . 7 ⊢ (1R +R -1R) = 0R
95	94	oveq2i 6029	. . . . . 6 ⊢ ((𝐹‘𝑚) +R (1R +R -1R)) = ((𝐹‘𝑚) +R 0R)
96		0idsr 7987	. . . . . . 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 2276	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘𝑚) +R (1R +R -1R)) = (𝐹‘𝑚))
99	90, 98	eqtrd 2264	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → (((𝐹‘𝑚) +R 1R) +R -1R) = (𝐹‘𝑚))
100	87, 99	breqtrd 4114	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ N) → ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
101	100	ralrimiva 2605	. 2 ⊢ (𝜑 → ∀𝑚 ∈ N ((𝐹‘1o) +R -1R) <R (𝐹‘𝑚))
102	1, 2, 101	caucvgsrlembnd 8021	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 1003   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511  ⟨cop 3672   class class class wbr 4088  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  1_oc1o 6575  [cec 6700  Ncnpi 7492   <_N clti 7495   ~_Q ceq 7499  1_Qc1q 7501  *_Qcrq 7504   <_Q cltq 7505  1_Pc1p 7512   +_P cpp 7513   ~_R cer 7516  Rcnr 7517  0_Rc0r 7518  1_Rc1r 7519  -1_Rcm1r 7520   +_R cplr 7521   <_R cltr 7523

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-iplp 7688  df-imp 7689  df-iltp 7690  df-enr 7946  df-nr 7947  df-plr 7948  df-mr 7949  df-ltr 7950  df-0r 7951  df-1r 7952  df-m1r 7953

This theorem is referenced by:  axcaucvglemres  8119