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Theorem caucvgsr 7248
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 7172 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 7247).

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 7243).

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 7172 to get a limit (see caucvgsrlemgt1 7241).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7241).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7246). (Contributed by Jim Kingdon, 20-Jun-2021.)

Hypotheses
Ref Expression
caucvgsr.f (𝜑𝐹:NR)
caucvgsr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Assertion
Ref Expression
caucvgsr (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +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 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgsr.f . 2 (𝜑𝐹:NR)
2 caucvgsr.cau . 2 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
3 breq1 3814 . . . . . . . . . . . . 13 (𝑛 = 1𝑜 → (𝑛 <N 𝑘 ↔ 1𝑜 <N 𝑘))
4 fveq2 5251 . . . . . . . . . . . . . . 15 (𝑛 = 1𝑜 → (𝐹𝑛) = (𝐹‘1𝑜))
5 opeq1 3596 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 1𝑜 → ⟨𝑛, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
65eceq1d 6256 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1𝑜 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
76fveq2d 5255 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1𝑜 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
87breq2d 3823 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1𝑜 → (𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
98abbidv 2200 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1𝑜 → {𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )})
107breq1d 3821 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1𝑜 → ((*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢))
1110abbidv 2200 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1𝑜 → {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢})
129, 11opeq12d 3604 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1𝑜 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
1312oveq1d 5604 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1𝑜 → (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P))
1413opeq1d 3602 . . . . . . . . . . . . . . . . 17 (𝑛 = 1𝑜 → ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩)
1514eceq1d 6256 . . . . . . . . . . . . . . . 16 (𝑛 = 1𝑜 → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
1615oveq2d 5605 . . . . . . . . . . . . . . 15 (𝑛 = 1𝑜 → ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
174, 16breq12d 3824 . . . . . . . . . . . . . 14 (𝑛 = 1𝑜 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
184, 15oveq12d 5607 . . . . . . . . . . . . . . 15 (𝑛 = 1𝑜 → ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
1918breq2d 3823 . . . . . . . . . . . . . 14 (𝑛 = 1𝑜 → ((𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
2017, 19anbi12d 457 . . . . . . . . . . . . 13 (𝑛 = 1𝑜 → (((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ↔ ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
213, 20imbi12d 232 . . . . . . . . . . . 12 (𝑛 = 1𝑜 → ((𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) ↔ (1𝑜 <N 𝑘 → ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
2221ralbidv 2374 . . . . . . . . . . 11 (𝑛 = 1𝑜 → (∀𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) ↔ ∀𝑘N (1𝑜 <N 𝑘 → ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
23 1pi 6775 . . . . . . . . . . . 12 1𝑜N
2423a1i 9 . . . . . . . . . . 11 (𝜑 → 1𝑜N)
2522, 2, 24rspcdva 2717 . . . . . . . . . 10 (𝜑 → ∀𝑘N (1𝑜 <N 𝑘 → ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
26 simpl 107 . . . . . . . . . . . 12 (((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
2726imim2i 12 . . . . . . . . . . 11 ((1𝑜 <N 𝑘 → ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → (1𝑜 <N 𝑘 → (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
2827ralimi 2432 . . . . . . . . . 10 (∀𝑘N (1𝑜 <N 𝑘 → ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹‘1𝑜) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → ∀𝑘N (1𝑜 <N 𝑘 → (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
2925, 28syl 14 . . . . . . . . 9 (𝜑 → ∀𝑘N (1𝑜 <N 𝑘 → (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
30 breq2 3815 . . . . . . . . . . 11 (𝑘 = 𝑚 → (1𝑜 <N 𝑘 ↔ 1𝑜 <N 𝑚))
31 fveq2 5251 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231oveq1d 5604 . . . . . . . . . . . 12 (𝑘 = 𝑚 → ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
3332breq2d 3823 . . . . . . . . . . 11 (𝑘 = 𝑚 → ((𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘1𝑜) <R ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
3430, 33imbi12d 232 . . . . . . . . . 10 (𝑘 = 𝑚 → ((1𝑜 <N 𝑘 → (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ↔ (1𝑜 <N 𝑚 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
3534rspcv 2708 . . . . . . . . 9 (𝑚N → (∀𝑘N (1𝑜 <N 𝑘 → (𝐹‘1𝑜) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (1𝑜 <N 𝑚 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
3629, 35mpan9 275 . . . . . . . 8 ((𝜑𝑚N) → (1𝑜 <N 𝑚 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
37 df-1nqqs 6811 . . . . . . . . . . . . . . . . . . . 20 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
3837fveq2i 5254 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
39 rec1nq 6855 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = 1Q
4038, 39eqtr3i 2105 . . . . . . . . . . . . . . . . . 18 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
4140breq2i 3819 . . . . . . . . . . . . . . . . 17 (𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q 1Q)
4241abbii 2198 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q 1Q}
4340breq1i 3818 . . . . . . . . . . . . . . . . 17 ((*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
4443abbii 2198 . . . . . . . . . . . . . . . 16 {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
4542, 44opeq12i 3601 . . . . . . . . . . . . . . 15 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46 df-i1p 6927 . . . . . . . . . . . . . . 15 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4745, 46eqtr4i 2106 . . . . . . . . . . . . . 14 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = 1P
4847oveq1i 5599 . . . . . . . . . . . . 13 (⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
4948opeq1i 3599 . . . . . . . . . . . 12 ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P
50 eceq1 6255 . . . . . . . . . . . 12 (⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩ → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R )
5149, 50ax-mp 7 . . . . . . . . . . 11 [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R
52 df-1r 7179 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
5351, 52eqtr4i 2106 . . . . . . . . . 10 [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
5453oveq2i 5600 . . . . . . . . 9 ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R 1R)
5554breq2i 3819 . . . . . . . 8 ((𝐹‘1𝑜) <R ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
5636, 55syl6ib 159 . . . . . . 7 ((𝜑𝑚N) → (1𝑜 <N 𝑚 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R)))
5756imp 122 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 <N 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
581adantr 270 . . . . . . . . . 10 ((𝜑𝑚N) → 𝐹:NR)
5923a1i 9 . . . . . . . . . 10 ((𝜑𝑚N) → 1𝑜N)
6058, 59ffvelrnd 5378 . . . . . . . . 9 ((𝜑𝑚N) → (𝐹‘1𝑜) ∈ R)
61 ltadd1sr 7223 . . . . . . . . 9 ((𝐹‘1𝑜) ∈ R → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
6260, 61syl 14 . . . . . . . 8 ((𝜑𝑚N) → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
6362adantr 270 . . . . . . 7 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
64 fveq2 5251 . . . . . . . . 9 (1𝑜 = 𝑚 → (𝐹‘1𝑜) = (𝐹𝑚))
6564oveq1d 5604 . . . . . . . 8 (1𝑜 = 𝑚 → ((𝐹‘1𝑜) +R 1R) = ((𝐹𝑚) +R 1R))
6665adantl 271 . . . . . . 7 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → ((𝐹‘1𝑜) +R 1R) = ((𝐹𝑚) +R 1R))
6763, 66breqtrd 3835 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
68 nlt1pig 6801 . . . . . . . . 9 (𝑚N → ¬ 𝑚 <N 1𝑜)
6968adantl 271 . . . . . . . 8 ((𝜑𝑚N) → ¬ 𝑚 <N 1𝑜)
7069pm2.21d 582 . . . . . . 7 ((𝜑𝑚N) → (𝑚 <N 1𝑜 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R)))
7170imp 122 . . . . . 6 (((𝜑𝑚N) ∧ 𝑚 <N 1𝑜) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
72 pitri3or 6782 . . . . . . . 8 ((1𝑜N𝑚N) → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7323, 72mpan 415 . . . . . . 7 (𝑚N → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7473adantl 271 . . . . . 6 ((𝜑𝑚N) → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7557, 67, 71, 74mpjao3dan 1239 . . . . 5 ((𝜑𝑚N) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
76 ltasrg 7217 . . . . . . 7 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
7776adantl 271 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
781ffvelrnda 5377 . . . . . . 7 ((𝜑𝑚N) → (𝐹𝑚) ∈ R)
79 1sr 7198 . . . . . . 7 1RR
80 addclsr 7200 . . . . . . 7 (((𝐹𝑚) ∈ R ∧ 1RR) → ((𝐹𝑚) +R 1R) ∈ R)
8178, 79, 80sylancl 404 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 1R) ∈ R)
82 m1r 7199 . . . . . . 7 -1RR
8382a1i 9 . . . . . 6 ((𝜑𝑚N) → -1RR)
84 addcomsrg 7202 . . . . . . 7 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8584adantl 271 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8677, 60, 81, 83, 85caovord2d 5747 . . . . 5 ((𝜑𝑚N) → ((𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R) ↔ ((𝐹‘1𝑜) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R)))
8775, 86mpbid 145 . . . 4 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R))
8879a1i 9 . . . . . 6 ((𝜑𝑚N) → 1RR)
89 addasssrg 7203 . . . . . 6 (((𝐹𝑚) ∈ R ∧ 1RR ∧ -1RR) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
9078, 88, 83, 89syl3anc 1170 . . . . 5 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
91 addcomsrg 7202 . . . . . . . . 9 ((1RR ∧ -1RR) → (1R +R -1R) = (-1R +R 1R))
9279, 82, 91mp2an 417 . . . . . . . 8 (1R +R -1R) = (-1R +R 1R)
93 m1p1sr 7207 . . . . . . . 8 (-1R +R 1R) = 0R
9492, 93eqtri 2103 . . . . . . 7 (1R +R -1R) = 0R
9594oveq2i 5600 . . . . . 6 ((𝐹𝑚) +R (1R +R -1R)) = ((𝐹𝑚) +R 0R)
96 0idsr 7214 . . . . . . 7 ((𝐹𝑚) ∈ R → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9778, 96syl 14 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9895, 97syl5eq 2127 . . . . 5 ((𝜑𝑚N) → ((𝐹𝑚) +R (1R +R -1R)) = (𝐹𝑚))
9990, 98eqtrd 2115 . . . 4 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = (𝐹𝑚))
10087, 99breqtrd 3835 . . 3 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
101100ralrimiva 2440 . 2 (𝜑 → ∀𝑚N ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
1021, 2, 101caucvgsrlembnd 7247 1 (𝜑 → ∃𝑦R𝑥R (0R <R 𝑥 → ∃𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹𝑘) +R 𝑥)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3o 919  w3a 920   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  cop 3425   class class class wbr 3811  wf 4963  cfv 4967  (class class class)co 5589  1𝑜c1o 6104  [cec 6218  Ncnpi 6732   <N clti 6735   ~Q ceq 6739  1Qc1q 6741  *Qcrq 6744   <Q cltq 6745  1Pc1p 6752   +P cpp 6753   ~R cer 6756  Rcnr 6757  0Rc0r 6758  1Rc1r 6759  -1Rcm1r 6760   +R cplr 6761   <R cltr 6763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-1o 6111  df-2o 6112  df-oadd 6115  df-omul 6116  df-er 6220  df-ec 6222  df-qs 6226  df-ni 6764  df-pli 6765  df-mi 6766  df-lti 6767  df-plpq 6804  df-mpq 6805  df-enq 6807  df-nqqs 6808  df-plqqs 6809  df-mqqs 6810  df-1nqqs 6811  df-rq 6812  df-ltnqqs 6813  df-enq0 6884  df-nq0 6885  df-0nq0 6886  df-plq0 6887  df-mq0 6888  df-inp 6926  df-i1p 6927  df-iplp 6928  df-imp 6929  df-iltp 6930  df-enr 7173  df-nr 7174  df-plr 7175  df-mr 7176  df-ltr 7177  df-0r 7178  df-1r 7179  df-m1r 7180
This theorem is referenced by:  axcaucvglemres  7335
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