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Theorem caucvgsr 6884
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 6808 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 6883).

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 6879).

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 6808 to get a limit (see caucvgsrlemgt1 6877).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6877).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6882). (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 6411 . . . . . . . . . . 11 1𝑜N
4 breq1 3767 . . . . . . . . . . . . . 14 (𝑛 = 1𝑜 → (𝑛 <N 𝑘 ↔ 1𝑜 <N 𝑘))
5 fveq2 5178 . . . . . . . . . . . . . . . 16 (𝑛 = 1𝑜 → (𝐹𝑛) = (𝐹‘1𝑜))
6 opeq1 3549 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 1𝑜 → ⟨𝑛, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
76eceq1d 6142 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 1𝑜 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
87fveq2d 5182 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1𝑜 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
98breq2d 3776 . . . . . . . . . . . . . . . . . . . . . 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 3774 . . . . . . . . . . . . . . . . . . . . . 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 3557 . . . . . . . . . . . . . . . . . . . 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 5527 . . . . . . . . . . . . . . . . . . 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 3555 . . . . . . . . . . . . . . . . . 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 6142 . . . . . . . . . . . . . . . . 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 5528 . . . . . . . . . . . . . . . 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 3777 . . . . . . . . . . . . . . 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 5530 . . . . . . . . . . . . . . . 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 3776 . . . . . . . . . . . . . . 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 2326 . . . . . . . . . . . 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 2652 . . . . . . . . . . 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 2384 . . . . . . . . . 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 3768 . . . . . . . . . . 11 (𝑘 = 𝑚 → (1𝑜 <N 𝑘 ↔ 1𝑜 <N 𝑚))
31 fveq2 5178 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231oveq1d 5527 . . . . . . . . . . . 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 3776 . . . . . . . . . . 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 2652 . . . . . . . . 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 6447 . . . . . . . . . . . . . . . . . . . 20 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
3837fveq2i 5181 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
39 rec1nq 6491 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = 1Q
4038, 39eqtr3i 2062 . . . . . . . . . . . . . . . . . 18 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
4140breq2i 3772 . . . . . . . . . . . . . . . . 17 (𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q 1Q)
4241abbii 2153 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q 1Q}
4340breq1i 3771 . . . . . . . . . . . . . . . . 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 3554 . . . . . . . . . . . . . . 15 ⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46 df-i1p 6563 . . . . . . . . . . . . . . 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 5522 . . . . . . . . . . . . 13 (⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
4948opeq1i 3552 . . . . . . . . . . . 12 ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P
50 eceq1 6141 . . . . . . . . . . . 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 6815 . . . . . . . . . . 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 5523 . . . . . . . . 9 ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R 1R)
5554breq2i 3772 . . . . . . . 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 5303 . . . . . . . . 9 ((𝜑𝑚N) → (𝐹‘1𝑜) ∈ R)
61 ltadd1sr 6859 . . . . . . . . 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 5178 . . . . . . . . 9 (1𝑜 = 𝑚 → (𝐹‘1𝑜) = (𝐹𝑚))
6564oveq1d 5527 . . . . . . . 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 3788 . . . . . 6 (((𝜑𝑚N) ∧ 1𝑜 = 𝑚) → (𝐹‘1𝑜) <R ((𝐹𝑚) +R 1R))
68 nlt1pig 6437 . . . . . . . . 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 6418 . . . . . . . 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 6853 . . . . . . 7 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
7776adantl 262 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
781ffvelrnda 5302 . . . . . . 7 ((𝜑𝑚N) → (𝐹𝑚) ∈ R)
79 1sr 6834 . . . . . . 7 1RR
80 addclsr 6836 . . . . . . 7 (((𝐹𝑚) ∈ R ∧ 1RR) → ((𝐹𝑚) +R 1R) ∈ R)
8178, 79, 80sylancl 392 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 1R) ∈ R)
82 m1r 6835 . . . . . . 7 -1RR
8382a1i 9 . . . . . 6 ((𝜑𝑚N) → -1RR)
84 addcomsrg 6838 . . . . . . 7 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8584adantl 262 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8677, 60, 81, 83, 85caovord2d 5670 . . . . 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 6839 . . . . . 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 6838 . . . . . . . . 9 ((1RR ∧ -1RR) → (1R +R -1R) = (-1R +R 1R))
9279, 82, 91mp2an 402 . . . . . . . 8 (1R +R -1R) = (-1R +R 1R)
93 m1p1sr 6843 . . . . . . . 8 (-1R +R 1R) = 0R
9492, 93eqtri 2060 . . . . . . 7 (1R +R -1R) = 0R
9594oveq2i 5523 . . . . . 6 ((𝐹𝑚) +R (1R +R -1R)) = ((𝐹𝑚) +R 0R)
96 0idsr 6850 . . . . . . 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 3788 . . 3 ((𝜑𝑚N) → ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
101100ralrimiva 2392 . 2 (𝜑 → ∀𝑚N ((𝐹‘1𝑜) +R -1R) <R (𝐹𝑚))
1021, 2, 101caucvgsrlembnd 6883 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 2306  wrex 2307  cop 3378   class class class wbr 3764  wf 4898  cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  1Qc1q 6377  *Qcrq 6380   <Q cltq 6381  1Pc1p 6388   +P cpp 6389   ~R cer 6392  Rcnr 6393  0Rc0r 6394  1Rc1r 6395  -1Rcm1r 6396   +R cplr 6397   <R cltr 6399
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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
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 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565  df-iltp 6566  df-enr 6809  df-nr 6810  df-plr 6811  df-mr 6812  df-ltr 6813  df-0r 6814  df-1r 6815  df-m1r 6816
This theorem is referenced by:  axcaucvglemres  6971
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