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Theorem caucvgsr 7444
 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 7368 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 7443). 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 7439). 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 7368 to get a limit (see caucvgsrlemgt1 7437). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 7437). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 7442). (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 3870 . . . . . . . . . . . . 13 (𝑛 = 1o → (𝑛 <N 𝑘 ↔ 1o <N 𝑘))
4 fveq2 5340 . . . . . . . . . . . . . . 15 (𝑛 = 1o → (𝐹𝑛) = (𝐹‘1o))
5 opeq1 3644 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 1o → ⟨𝑛, 1o⟩ = ⟨1o, 1o⟩)
65eceq1d 6368 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1o → [⟨𝑛, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
76fveq2d 5344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1o → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
87breq2d 3879 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1o → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
98abbidv 2212 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1o → {𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
107breq1d 3877 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1o → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢))
1110abbidv 2212 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1o → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢})
129, 11opeq12d 3652 . . . . . . . . . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . . 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 3650 . . . . . . . . . . . . . . . . 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 6368 . . . . . . . . . . . . . . . 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 5706 . . . . . . . . . . . . . . 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 3880 . . . . . . . . . . . . . 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 5708 . . . . . . . . . . . . . . 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 3879 . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . 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 2391 . . . . . . . . . . 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 6971 . . . . . . . . . . . 12 1oN
2423a1i 9 . . . . . . . . . . 11 (𝜑 → 1oN)
2522, 2, 24rspcdva 2741 . . . . . . . . . 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 2449 . . . . . . . . . 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 3871 . . . . . . . . . . 11 (𝑘 = 𝑚 → (1o <N 𝑘 ↔ 1o <N 𝑚))
31 fveq2 5340 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231oveq1d 5705 . . . . . . . . . . . 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 3879 . . . . . . . . . . 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 2732 . . . . . . . . 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 276 . . . . . . . 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 7007 . . . . . . . . . . . . . . . . . . . 20 1Q = [⟨1o, 1o⟩] ~Q
3837fveq2i 5343 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
39 rec1nq 7051 . . . . . . . . . . . . . . . . . . 19 (*Q‘1Q) = 1Q
4038, 39eqtr3i 2117 . . . . . . . . . . . . . . . . . 18 (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
4140breq2i 3875 . . . . . . . . . . . . . . . . 17 (𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑙 <Q 1Q)
4241abbii 2210 . . . . . . . . . . . . . . . 16 {𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑙𝑙 <Q 1Q}
4340breq1i 3874 . . . . . . . . . . . . . . . . 17 ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢 ↔ 1Q <Q 𝑢)
4443abbii 2210 . . . . . . . . . . . . . . . 16 {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢}
4542, 44opeq12i 3649 . . . . . . . . . . . . . . 15 ⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
46 df-i1p 7123 . . . . . . . . . . . . . . 15 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4745, 46eqtr4i 2118 . . . . . . . . . . . . . 14 ⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ = 1P
4847oveq1i 5700 . . . . . . . . . . . . 13 (⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P) = (1P +P 1P)
4948opeq1i 3647 . . . . . . . . . . . 12 ⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P
50 eceq1 6367 . . . . . . . . . . . 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 7375 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
5351, 52eqtr4i 2118 . . . . . . . . . 10 [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R
5453oveq2i 5701 . . . . . . . . 9 ((𝐹𝑚) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑚) +R 1R)
5554breq2i 3875 . . . . . . . 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 271 . . . . . . . . . 10 ((𝜑𝑚N) → 𝐹:NR)
5923a1i 9 . . . . . . . . . 10 ((𝜑𝑚N) → 1oN)
6058, 59ffvelrnd 5474 . . . . . . . . 9 ((𝜑𝑚N) → (𝐹‘1o) ∈ R)
61 ltadd1sr 7419 . . . . . . . . 9 ((𝐹‘1o) ∈ R → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
6260, 61syl 14 . . . . . . . 8 ((𝜑𝑚N) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
6362adantr 271 . . . . . . 7 (((𝜑𝑚N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹‘1o) +R 1R))
64 fveq2 5340 . . . . . . . . 9 (1o = 𝑚 → (𝐹‘1o) = (𝐹𝑚))
6564oveq1d 5705 . . . . . . . 8 (1o = 𝑚 → ((𝐹‘1o) +R 1R) = ((𝐹𝑚) +R 1R))
6665adantl 272 . . . . . . 7 (((𝜑𝑚N) ∧ 1o = 𝑚) → ((𝐹‘1o) +R 1R) = ((𝐹𝑚) +R 1R))
6763, 66breqtrd 3891 . . . . . 6 (((𝜑𝑚N) ∧ 1o = 𝑚) → (𝐹‘1o) <R ((𝐹𝑚) +R 1R))
68 nlt1pig 6997 . . . . . . . . 9 (𝑚N → ¬ 𝑚 <N 1o)
6968adantl 272 . . . . . . . 8 ((𝜑𝑚N) → ¬ 𝑚 <N 1o)
7069pm2.21d 587 . . . . . . 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 6978 . . . . . . . 8 ((1oN𝑚N) → (1o <N 𝑚 ∨ 1o = 𝑚𝑚 <N 1o))
7323, 72mpan 416 . . . . . . 7 (𝑚N → (1o <N 𝑚 ∨ 1o = 𝑚𝑚 <N 1o))
7473adantl 272 . . . . . 6 ((𝜑𝑚N) → (1o <N 𝑚 ∨ 1o = 𝑚𝑚 <N 1o))
7557, 67, 71, 74mpjao3dan 1250 . . . . 5 ((𝜑𝑚N) → (𝐹‘1o) <R ((𝐹𝑚) +R 1R))
76 ltasrg 7413 . . . . . . 7 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
7776adantl 272 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
781ffvelrnda 5473 . . . . . . 7 ((𝜑𝑚N) → (𝐹𝑚) ∈ R)
79 1sr 7394 . . . . . . 7 1RR
80 addclsr 7396 . . . . . . 7 (((𝐹𝑚) ∈ R ∧ 1RR) → ((𝐹𝑚) +R 1R) ∈ R)
8178, 79, 80sylancl 405 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 1R) ∈ R)
82 m1r 7395 . . . . . . 7 -1RR
8382a1i 9 . . . . . 6 ((𝜑𝑚N) → -1RR)
84 addcomsrg 7398 . . . . . . 7 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8584adantl 272 . . . . . 6 (((𝜑𝑚N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
8677, 60, 81, 83, 85caovord2d 5852 . . . . 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 7399 . . . . . 6 (((𝐹𝑚) ∈ R ∧ 1RR ∧ -1RR) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
9078, 88, 83, 89syl3anc 1181 . . . . 5 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = ((𝐹𝑚) +R (1R +R -1R)))
91 addcomsrg 7398 . . . . . . . . 9 ((1RR ∧ -1RR) → (1R +R -1R) = (-1R +R 1R))
9279, 82, 91mp2an 418 . . . . . . . 8 (1R +R -1R) = (-1R +R 1R)
93 m1p1sr 7403 . . . . . . . 8 (-1R +R 1R) = 0R
9492, 93eqtri 2115 . . . . . . 7 (1R +R -1R) = 0R
9594oveq2i 5701 . . . . . 6 ((𝐹𝑚) +R (1R +R -1R)) = ((𝐹𝑚) +R 0R)
96 0idsr 7410 . . . . . . 7 ((𝐹𝑚) ∈ R → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9778, 96syl 14 . . . . . 6 ((𝜑𝑚N) → ((𝐹𝑚) +R 0R) = (𝐹𝑚))
9895, 97syl5eq 2139 . . . . 5 ((𝜑𝑚N) → ((𝐹𝑚) +R (1R +R -1R)) = (𝐹𝑚))
9990, 98eqtrd 2127 . . . 4 ((𝜑𝑚N) → (((𝐹𝑚) +R 1R) +R -1R) = (𝐹𝑚))
10087, 99breqtrd 3891 . . 3 ((𝜑𝑚N) → ((𝐹‘1o) +R -1R) <R (𝐹𝑚))
101100ralrimiva 2458 . 2 (𝜑 → ∀𝑚N ((𝐹‘1o) +R -1R) <R (𝐹𝑚))
1021, 2, 101caucvgsrlembnd 7443 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 926   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  {cab 2081  ∀wral 2370  ∃wrex 2371  ⟨cop 3469   class class class wbr 3867  ⟶wf 5045  ‘cfv 5049  (class class class)co 5690  1oc1o 6212  [cec 6330  Ncnpi 6928
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