ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgsr GIF version

Theorem caucvgsr 7603
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 7513 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 7602).

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 7598).

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 7513 to get a limit (see caucvgsrlemgt1 7596).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7596).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7601). (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 3927 . . . . . . . . . . . . 13 (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘))
4 fveq2 5414 . . . . . . . . . . . . . . 15 (𝑛 = 1o → (𝐹𝑛) = (𝐹‘1o))
5 opeq1 3700 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 1o → ⟨𝑛, 1o⟩ = ⟨1o, 1o⟩)
65eceq1d 6458 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1o → [⟨𝑛, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
76fveq2d 5418 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1o → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
87breq2d 3936 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1o → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
98abbidv 2255 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1o → {𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
107breq1d 3934 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1o → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢))
1110abbidv 2255 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1o → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢})
129, 11opeq12d 3708 . . . . . . . . . . . . . . . . . . 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 5782 . . . . . . . . . . . . . . . . . 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 3706 . . . . . . . . . . . . . . . . 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 6458 . . . . . . . . . . . . . . . 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 5783 . . . . . . . . . . . . . . 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 3937 . . . . . . . . . . . . . 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 5785 . . . . . . . . . . . . . . 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 3936 . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . 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 233 . . . . . . . . . . . 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 2435 . . . . . . . . . . 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 7116 . . . . . . . . . . . 12 1oN
2423a1i 9 . . . . . . . . . . 11 (𝜑 → 1oN)
2522, 2, 24rspcdva 2789 . . . . . . . . . 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 108 . . . . . . . . . . . 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 2493 . . . . . . . . . 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 3928 . . . . . . . . . . 11 (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚))
31 fveq2 5414 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231oveq1d 5782 . . . . . . . . . . . 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 3936 . . . . . . . . . . 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 233 . . . . . . . . . 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 2780 . . . . . . . . 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 279 . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . 20 1Q = [⟨1o, 1o⟩] ~Q
3837fveq2i 5417 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
39 rec1nq 7196 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = 1Q
4038, 39eqtr3i 2160 . . . . . . . . . . . . . . . . . 18 (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
4140breq2i 3932 . . . . . . . . . . . . . . . . 17 (𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑙 <Q 1Q)
4241abbii 2253 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑙𝑙 <Q 1Q}
4340breq1i 3931 . . . . . . . . . . . . . . . . 17 ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
4443abbii 2253 . . . . . . . . . . . . . . . 16 {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
4542, 44opeq12i 3705 . . . . . . . . . . . . . . 15 ⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46 df-i1p 7268 . . . . . . . . . . . . . . 15 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4745, 46eqtr4i 2161 . . . . . . . . . . . . . 14 ⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = 1P
4847oveq1i 5777 . . . . . . . . . . . . 13 (⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
4948opeq1i 3703 . . . . . . . . . . . 12 ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P
50 eceq1 6457 . . . . . . . . . . . 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 5 . . . . . . . . . . 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 7533 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
5351, 52eqtr4i 2161 . . . . . . . . . 10 [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
5453oveq2i 5778 . . . . . . . . 9 ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R 1R)
5554breq2i 3932 . . . . . . . 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 160 . . . . . . 7 ((𝜑𝑚N) → (1o <N 𝑚 → (𝐹‘1o) <R ((𝐹𝑚) +R 1R)))
5756imp 123 . . . . . 6 (((𝜑𝑚N) ∧ 1o <N 𝑚) → (𝐹‘1o) <R ((𝐹𝑚) +R 1R))
581adantr 274 . . . . . . . . . 10 ((𝜑𝑚N) → 𝐹:NR)
5923a1i 9 . . . . . . . . . 10 ((𝜑𝑚N) → 1oN)
6058, 59ffvelrnd 5549 . . . . . . . . 9 ((𝜑𝑚N) → (𝐹‘1o) ∈ R)
61 ltadd1sr 7577 . . . . . . . . 9 ((𝐹‘1o) ∈ R → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
6260, 61syl 14 . . . . . . . 8 ((𝜑𝑚N) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
6362adantr 274 . . . . . . 7 (((𝜑𝑚N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
64 fveq2 5414 . . . . . . . . 9 (1o = 𝑚 → (𝐹‘1o) = (𝐹𝑚))
6564oveq1d 5782 . . . . . . . 8 (1o = 𝑚 → ((𝐹‘1o) +R 1R) = ((𝐹𝑚) +R 1R))
6665adantl 275 . . . . . . 7 (((𝜑𝑚N) ∧ 1o = 𝑚) → ((𝐹‘1o) +R 1R) = ((𝐹𝑚) +R 1R))
6763, 66breqtrd 3949 . . . . . 6 (((𝜑𝑚N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹𝑚) +R 1R))
68 nlt1pig 7142 . . . . . . . . 9 (𝑚N → ¬ 𝑚 <N 1o)
6968adantl 275 . . . . . . . 8 ((𝜑𝑚N) → ¬ 𝑚 <N 1o)
7069pm2.21d 608 . . . . . . 7 ((𝜑𝑚N) → (𝑚 <N 1o → (𝐹‘1o) <R ((𝐹𝑚) +R 1R)))
7170imp 123 . . . . . 6 (((𝜑𝑚N) ∧ 𝑚 <N 1o) → (𝐹‘1o) <R ((𝐹𝑚) +R 1R))
72 pitri3or 7123 . . . . . . . 8 ((1oN𝑚N) → (1o <N 𝑚 ∨ 1o = 𝑚𝑚 <N 1o))
7323, 72mpan 420 . . . . . . 7 (𝑚N → (1o <N 𝑚 ∨ 1o = 𝑚𝑚 <N 1o))
7473adantl 275 . . . . . 6 ((𝜑𝑚N) → (1o <N 𝑚 ∨ 1o = 𝑚𝑚 <N 1o))
7557, 67, 71, 74mpjao3dan 1285 . . . . 5 ((𝜑𝑚N) → (𝐹‘1o) <R ((𝐹𝑚) +R 1R))
76 ltasrg 7571 . . . . . . 7 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
7776adantl 275 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
781ffvelrnda 5548 . . . . . . 7 ((𝜑𝑚N) → (𝐹𝑚) ∈ R)
79 1sr 7552 . . . . . . 7 1RR
80 addclsr 7554 . . . . . . 7 (((𝐹𝑚) ∈ R ∧ 1RR) → ((𝐹𝑚) +R 1R) ∈ R)
8178, 79, 80sylancl 409 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 1R) ∈ R)
82 m1r 7553 . . . . . . 7 -1RR
8382a1i 9 . . . . . 6 ((𝜑𝑚N) → -1RR)
84 addcomsrg 7556 . . . . . . 7 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8584adantl 275 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8677, 60, 81, 83, 85caovord2d 5933 . . . . 5 ((𝜑𝑚N) → ((𝐹‘1o) <R ((𝐹𝑚) +R 1R) ↔ ((𝐹‘1o) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R)))
8775, 86mpbid 146 . . . 4 ((𝜑𝑚N) → ((𝐹‘1o) +R -1R) <R (((𝐹𝑚) +R 1R) +R -1R))
8879a1i 9 . . . . . 6 ((𝜑𝑚N) → 1RR)
89 addasssrg 7557 . . . . . 6 (((𝐹𝑚) ∈ R ∧ 1RR ∧ -1RR) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
9078, 88, 83, 89syl3anc 1216 . . . . 5 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
91 addcomsrg 7556 . . . . . . . . 9 ((1RR ∧ -1RR) → (1R +R -1R) = (-1R +R 1R))
9279, 82, 91mp2an 422 . . . . . . . 8 (1R +R -1R) = (-1R +R 1R)
93 m1p1sr 7561 . . . . . . . 8 (-1R +R 1R) = 0R
9492, 93eqtri 2158 . . . . . . 7 (1R +R -1R) = 0R
9594oveq2i 5778 . . . . . 6 ((𝐹𝑚) +R (1R +R -1R)) = ((𝐹𝑚) +R 0R)
96 0idsr 7568 . . . . . . 7 ((𝐹𝑚) ∈ R → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9778, 96syl 14 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9895, 97syl5eq 2182 . . . . 5 ((𝜑𝑚N) → ((𝐹𝑚) +R (1R +R -1R)) = (𝐹𝑚))
9990, 98eqtrd 2170 . . . 4 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = (𝐹𝑚))
10087, 99breqtrd 3949 . . 3 ((𝜑𝑚N) → ((𝐹‘1o) +R -1R) <R (𝐹𝑚))
101100ralrimiva 2503 . 2 (𝜑 → ∀𝑚N ((𝐹‘1o) +R -1R) <R (𝐹𝑚))
1021, 2, 101caucvgsrlembnd 7602 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 103  wb 104  w3o 961  w3a 962   = wceq 1331  wcel 1480  {cab 2123  wral 2414  wrex 2415  cop 3525   class class class wbr 3924  wf 5114  cfv 5118  (class class class)co 5767  1oc1o 6299  [cec 6420  Ncnpi 7073   <N clti 7076   ~Q ceq 7080  1Qc1q 7082  *Qcrq 7085   <Q cltq 7086  1Pc1p 7093   +P cpp 7094   ~R cer 7097  Rcnr 7098  0Rc0r 7099  1Rc1r 7100  -1Rcm1r 7101   +R cplr 7102   <R cltr 7104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-imp 7270  df-iltp 7271  df-enr 7527  df-nr 7528  df-plr 7529  df-mr 7530  df-ltr 7531  df-0r 7532  df-1r 7533  df-m1r 7534
This theorem is referenced by:  axcaucvglemres  7700
  Copyright terms: Public domain W3C validator