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Theorem caucvgsr 6843
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 6767 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 6842).

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 6838).

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 6767 to get a limit (see caucvgsrlemgt1 6836).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6836).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6841). (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 6370 . . . . . . . . . . 11 1𝑜N
4 breq1 3764 . . . . . . . . . . . . . 14 (𝑛 = 1𝑜 → (𝑛 <N 𝑘 ↔ 1𝑜 <N 𝑘))
5 fveq2 5141 . . . . . . . . . . . . . . . 16 (𝑛 = 1𝑜 → (𝐹𝑛) = (𝐹‘1𝑜))
6 opeq1 3546 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 1𝑜 → ⟨𝑛, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
76eceq1d 6105 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 1𝑜 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
87fveq2d 5145 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1𝑜 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
98breq2d 3773 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1𝑜 → (𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
109abbidv 2155 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1𝑜 → {𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )})
118breq1d 3771 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1𝑜 → ((*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢))
1211abbidv 2155 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1𝑜 → {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢})
1310, 12opeq12d 3554 . . . . . . . . . . . . . . . . . . . 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 5490 . . . . . . . . . . . . . . . . . . 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 3552 . . . . . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . . . . 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 5491 . . . . . . . . . . . . . . . 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 3774 . . . . . . . . . . . . . . 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 5493 . . . . . . . . . . . . . . . 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 3773 . . . . . . . . . . . . . . 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 442 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 2323 . . . . . . . . . . . 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 2649 . . . . . . . . . . 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 59 . . . . . . . . . 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 102 . . . . . . . . . . . 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 2381 . . . . . . . . . 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 3765 . . . . . . . . . . 11 (𝑘 = 𝑚 → (1𝑜 <N 𝑘 ↔ 1𝑜 <N 𝑚))
31 fveq2 5141 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231oveq1d 5490 . . . . . . . . . . . 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 3773 . . . . . . . . . . 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 223 . . . . . . . . . 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 2649 . . . . . . . . 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 265 . . . . . . . 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 6406 . . . . . . . . . . . . . . . . . . . 20 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
3837fveq2i 5144 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
39 rec1nq 6450 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = 1Q
4038, 39eqtr3i 2062 . . . . . . . . . . . . . . . . . 18 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
4140breq2i 3769 . . . . . . . . . . . . . . . . 17 (𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q 1Q)
4241abbii 2153 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q 1Q}
4340breq1i 3768 . . . . . . . . . . . . . . . . 17 ((*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
4443abbii 2153 . . . . . . . . . . . . . . . 16 {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
4542, 44opeq12i 3551 . . . . . . . . . . . . . . 15 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46 df-i1p 6522 . . . . . . . . . . . . . . 15 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4745, 46eqtr4i 2063 . . . . . . . . . . . . . 14 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = 1P
4847oveq1i 5485 . . . . . . . . . . . . 13 (⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
4948opeq1i 3549 . . . . . . . . . . . 12 ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P
50 eceq1 6104 . . . . . . . . . . . 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 6774 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
5351, 52eqtr4i 2063 . . . . . . . . . 10 [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
5453oveq2i 5486 . . . . . . . . 9 ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R 1R)
5554breq2i 3769 . . . . . . . 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 150 . . . . . . 7 ((𝜑𝑚N) → (1𝑜 <N 𝑚 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R)))
5756imp 115 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 <N 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
581adantr 261 . . . . . . . . . 10 ((𝜑𝑚N) → 𝐹:NR)
593a1i 9 . . . . . . . . . 10 ((𝜑𝑚N) → 1𝑜N)
6058, 59ffvelrnd 5266 . . . . . . . . 9 ((𝜑𝑚N) → (𝐹‘1𝑜) ∈ R)
61 ltadd1sr 6818 . . . . . . . . 9 ((𝐹‘1𝑜) ∈ R → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
6260, 61syl 14 . . . . . . . 8 ((𝜑𝑚N) → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
6362adantr 261 . . . . . . 7 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹‘1𝑜) +R 1R))
64 fveq2 5141 . . . . . . . . 9 (1𝑜 = 𝑚 → (𝐹‘1𝑜) = (𝐹𝑚))
6564oveq1d 5490 . . . . . . . 8 (1𝑜 = 𝑚 → ((𝐹‘1𝑜) +R 1R) = ((𝐹𝑚) +R 1R))
6665adantl 262 . . . . . . 7 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → ((𝐹‘1𝑜) +R 1R) = ((𝐹𝑚) +R 1R))
6763, 66breqtrd 3785 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
68 nlt1pig 6396 . . . . . . . . 9 (𝑚N → ¬ 𝑚 <N 1𝑜)
6968adantl 262 . . . . . . . 8 ((𝜑𝑚N) → ¬ 𝑚 <N 1𝑜)
7069pm2.21d 549 . . . . . . 7 ((𝜑𝑚N) → (𝑚 <N 1𝑜 → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R)))
7170imp 115 . . . . . 6 (((𝜑𝑚N) ∧ 𝑚 <N 1𝑜) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
72 pitri3or 6377 . . . . . . . 8 ((1𝑜N𝑚N) → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
733, 72mpan 400 . . . . . . 7 (𝑚N → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7473adantl 262 . . . . . 6 ((𝜑𝑚N) → (1𝑜 <N 𝑚 ∨ 1𝑜 = 𝑚𝑚 <N 1𝑜))
7557, 67, 71, 74mpjao3dan 1202 . . . . 5 ((𝜑𝑚N) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
76 ltasrg 6812 . . . . . . 7 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
7776adantl 262 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
781ffvelrnda 5265 . . . . . . 7 ((𝜑𝑚N) → (𝐹𝑚) ∈ R)
79 1sr 6793 . . . . . . 7 1RR
80 addclsr 6795 . . . . . . 7 (((𝐹𝑚) ∈ R ∧ 1RR) → ((𝐹𝑚) +R 1R) ∈ R)
8178, 79, 80sylancl 392 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 1R) ∈ R)
82 m1r 6794 . . . . . . 7 -1RR
8382a1i 9 . . . . . 6 ((𝜑𝑚N) → -1RR)
84 addcomsrg 6797 . . . . . . 7 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8584adantl 262 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8677, 60, 81, 83, 85caovord2d 5633 . . . . 5 ((𝜑𝑚N) → ((𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R) ↔ ((𝐹‘1𝑜) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R)))
8775, 86mpbid 135 . . . 4 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R))
8879a1i 9 . . . . . 6 ((𝜑𝑚N) → 1RR)
89 addasssrg 6798 . . . . . 6 (((𝐹𝑚) ∈ R ∧ 1RR ∧ -1RR) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
9078, 88, 83, 89syl3anc 1135 . . . . 5 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
91 addcomsrg 6797 . . . . . . . . 9 ((1RR ∧ -1RR) → (1R +R -1R) = (-1R +R 1R))
9279, 82, 91mp2an 402 . . . . . . . 8 (1R +R -1R) = (-1R +R 1R)
93 m1p1sr 6802 . . . . . . . 8 (-1R +R 1R) = 0R
9492, 93eqtri 2060 . . . . . . 7 (1R +R -1R) = 0R
9594oveq2i 5486 . . . . . 6 ((𝐹𝑚) +R (1R +R -1R)) = ((𝐹𝑚) +R 0R)
96 0idsr 6809 . . . . . . 7 ((𝐹𝑚) ∈ R → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9778, 96syl 14 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9895, 97syl5eq 2084 . . . . 5 ((𝜑𝑚N) → ((𝐹𝑚) +R (1R +R -1R)) = (𝐹𝑚))
9990, 98eqtrd 2072 . . . 4 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = (𝐹𝑚))
10087, 99breqtrd 3785 . . 3 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
101100ralrimiva 2389 . 2 (𝜑 → ∀𝑚N ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
1021, 2, 101caucvgsrlembnd 6842 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 97  wb 98  w3o 884  w3a 885   = wceq 1243  wcel 1393  {cab 2026  wral 2303  wrex 2304  cop 3375   class class class wbr 3761  wf 4861  cfv 4865  (class class class)co 5475  1𝑜c1o 5957  [cec 6067  Ncnpi 6327   <N clti 6330   ~Q ceq 6334  1Qc1q 6336  *Qcrq 6339   <Q cltq 6340  1Pc1p 6347   +P cpp 6348   ~R cer 6351  Rcnr 6352  0Rc0r 6353  1Rc1r 6354  -1Rcm1r 6355   +R cplr 6356   <R cltr 6358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rmo 2311  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-riota 5431  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-i1p 6522  df-iplp 6523  df-imp 6524  df-iltp 6525  df-enr 6768  df-nr 6769  df-plr 6770  df-mr 6771  df-ltr 6772  df-0r 6773  df-1r 6774  df-m1r 6775
This theorem is referenced by:  axcaucvglemres  6930
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