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Theorem caucvgsr 6944
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 6868 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 6943).

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 6939).

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 6868 to get a limit (see caucvgsrlemgt1 6937).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6937).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6942). (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 1pi 6471 . . . . . . . . . . 11 1𝑜N
4 breq1 3795 . . . . . . . . . . . . . 14 (𝑛 = 1𝑜 → (𝑛 <N 𝑘 ↔ 1𝑜 <N 𝑘))
5 fveq2 5206 . . . . . . . . . . . . . . . 16 (𝑛 = 1𝑜 → (𝐹𝑛) = (𝐹‘1𝑜))
6 opeq1 3577 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 1𝑜 → ⟨𝑛, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
76eceq1d 6173 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 1𝑜 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
87fveq2d 5210 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1𝑜 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
98breq2d 3804 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1𝑜 → (𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
109abbidv 2171 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1𝑜 → {𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )})
118breq1d 3802 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1𝑜 → ((*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢))
1211abbidv 2171 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1𝑜 → {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢})
1310, 12opeq12d 3585 . . . . . . . . . . . . . . . . . . . 20 (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}⟩)
1413oveq1d 5555 . . . . . . . . . . . . . . . . . . 19 (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))
1514opeq1d 3583 . . . . . . . . . . . . . . . . . 18 (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⟩)
1615eceq1d 6173 . . . . . . . . . . . . . . . . 17 (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 )
1716oveq2d 5556 . . . . . . . . . . . . . . . 16 (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 ))
185, 17breq12d 3805 . . . . . . . . . . . . . . 15 (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 )))
195, 16oveq12d 5558 . . . . . . . . . . . . . . . 16 (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 ))
2019breq2d 3804 . . . . . . . . . . . . . . 15 (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 )))
2118, 20anbi12d 450 . . . . . . . . . . . . . 14 (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 ))))
224, 21imbi12d 227 . . . . . . . . . . . . 13 (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 )))))
2322ralbidv 2343 . . . . . . . . . . . 12 (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 )))))
2423rspcv 2669 . . . . . . . . . . 11 (1𝑜N → (∀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 ))) → ∀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 )))))
253, 2, 24mpsyl 63 . . . . . . . . . 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 106 . . . . . . . . . . . 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 2401 . . . . . . . . . 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 3796 . . . . . . . . . . 11 (𝑘 = 𝑚 → (1𝑜 <N 𝑘 ↔ 1𝑜 <N 𝑚))
31 fveq2 5206 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231oveq1d 5555 . . . . . . . . . . . 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 3804 . . . . . . . . . . 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 227 . . . . . . . . . 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 2669 . . . . . . . . 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 269 . . . . . . . 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 6507 . . . . . . . . . . . . . . . . . . . 20 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
3837fveq2i 5209 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
39 rec1nq 6551 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = 1Q
4038, 39eqtr3i 2078 . . . . . . . . . . . . . . . . . 18 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
4140breq2i 3800 . . . . . . . . . . . . . . . . 17 (𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q 1Q)
4241abbii 2169 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q 1Q}
4340breq1i 3799 . . . . . . . . . . . . . . . . 17 ((*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
4443abbii 2169 . . . . . . . . . . . . . . . 16 {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
4542, 44opeq12i 3582 . . . . . . . . . . . . . . 15 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46 df-i1p 6623 . . . . . . . . . . . . . . 15 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4745, 46eqtr4i 2079 . . . . . . . . . . . . . 14 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = 1P
4847oveq1i 5550 . . . . . . . . . . . . 13 (⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
4948opeq1i 3580 . . . . . . . . . . . 12 ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P
50 eceq1 6172 . . . . . . . . . . . 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 6875 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
5351, 52eqtr4i 2079 . . . . . . . . . 10 [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
5453oveq2i 5551 . . . . . . . . 9 ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R 1R)
5554breq2i 3800 . . . . . . . 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 154 . . . . . . 7 ((𝜑𝑚N) → (1𝑜 <N 𝑚 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R)))
5756imp 119 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 <N 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
581adantr 265 . . . . . . . . . 10 ((𝜑𝑚N) → 𝐹:NR)
593a1i 9 . . . . . . . . . 10 ((𝜑𝑚N) → 1𝑜N)
6058, 59ffvelrnd 5331 . . . . . . . . 9 ((𝜑𝑚N) → (𝐹‘1𝑜) ∈ R)
61 ltadd1sr 6919 . . . . . . . . 9 ((𝐹‘1𝑜) ∈ R → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
6260, 61syl 14 . . . . . . . 8 ((𝜑𝑚N) → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
6362adantr 265 . . . . . . 7 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
64 fveq2 5206 . . . . . . . . 9 (1𝑜 = 𝑚 → (𝐹‘1𝑜) = (𝐹𝑚))
6564oveq1d 5555 . . . . . . . 8 (1𝑜 = 𝑚 → ((𝐹‘1𝑜) +R 1R) = ((𝐹𝑚) +R 1R))
6665adantl 266 . . . . . . 7 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → ((𝐹‘1𝑜) +R 1R) = ((𝐹𝑚) +R 1R))
6763, 66breqtrd 3816 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
68 nlt1pig 6497 . . . . . . . . 9 (𝑚N → ¬ 𝑚 <N 1𝑜)
6968adantl 266 . . . . . . . 8 ((𝜑𝑚N) → ¬ 𝑚 <N 1𝑜)
7069pm2.21d 559 . . . . . . 7 ((𝜑𝑚N) → (𝑚 <N 1𝑜 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R)))
7170imp 119 . . . . . 6 (((𝜑𝑚N) ∧ 𝑚 <N 1𝑜) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
72 pitri3or 6478 . . . . . . . 8 ((1𝑜N𝑚N) → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
733, 72mpan 408 . . . . . . 7 (𝑚N → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7473adantl 266 . . . . . 6 ((𝜑𝑚N) → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7557, 67, 71, 74mpjao3dan 1213 . . . . 5 ((𝜑𝑚N) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
76 ltasrg 6913 . . . . . . 7 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
7776adantl 266 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
781ffvelrnda 5330 . . . . . . 7 ((𝜑𝑚N) → (𝐹𝑚) ∈ R)
79 1sr 6894 . . . . . . 7 1RR
80 addclsr 6896 . . . . . . 7 (((𝐹𝑚) ∈ R ∧ 1RR) → ((𝐹𝑚) +R 1R) ∈ R)
8178, 79, 80sylancl 398 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 1R) ∈ R)
82 m1r 6895 . . . . . . 7 -1RR
8382a1i 9 . . . . . 6 ((𝜑𝑚N) → -1RR)
84 addcomsrg 6898 . . . . . . 7 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8584adantl 266 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8677, 60, 81, 83, 85caovord2d 5698 . . . . 5 ((𝜑𝑚N) → ((𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R) ↔ ((𝐹‘1𝑜) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R)))
8775, 86mpbid 139 . . . 4 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R))
8879a1i 9 . . . . . 6 ((𝜑𝑚N) → 1RR)
89 addasssrg 6899 . . . . . 6 (((𝐹𝑚) ∈ R ∧ 1RR ∧ -1RR) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
9078, 88, 83, 89syl3anc 1146 . . . . 5 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
91 addcomsrg 6898 . . . . . . . . 9 ((1RR ∧ -1RR) → (1R +R -1R) = (-1R +R 1R))
9279, 82, 91mp2an 410 . . . . . . . 8 (1R +R -1R) = (-1R +R 1R)
93 m1p1sr 6903 . . . . . . . 8 (-1R +R 1R) = 0R
9492, 93eqtri 2076 . . . . . . 7 (1R +R -1R) = 0R
9594oveq2i 5551 . . . . . 6 ((𝐹𝑚) +R (1R +R -1R)) = ((𝐹𝑚) +R 0R)
96 0idsr 6910 . . . . . . 7 ((𝐹𝑚) ∈ R → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9778, 96syl 14 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9895, 97syl5eq 2100 . . . . 5 ((𝜑𝑚N) → ((𝐹𝑚) +R (1R +R -1R)) = (𝐹𝑚))
9990, 98eqtrd 2088 . . . 4 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = (𝐹𝑚))
10087, 99breqtrd 3816 . . 3 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
101100ralrimiva 2409 . 2 (𝜑 → ∀𝑚N ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
1021, 2, 101caucvgsrlembnd 6943 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 101  wb 102  w3o 895  w3a 896   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  cop 3406   class class class wbr 3792  wf 4926  cfv 4930  (class class class)co 5540  1𝑜c1o 6025  [cec 6135  Ncnpi 6428   <N clti 6431   ~Q ceq 6435  1Qc1q 6437  *Qcrq 6440   <Q cltq 6441  1Pc1p 6448   +P cpp 6449   ~R cer 6452  Rcnr 6453  0Rc0r 6454  1Rc1r 6455  -1Rcm1r 6456   +R cplr 6457   <R cltr 6459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-imp 6625  df-iltp 6626  df-enr 6869  df-nr 6870  df-plr 6871  df-mr 6872  df-ltr 6873  df-0r 6874  df-1r 6875  df-m1r 6876
This theorem is referenced by:  axcaucvglemres  7031
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