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Theorem caucvgsrlemcau 6934
 Description: Lemma for caucvgsr 6943. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-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 ))))
caucvgsrlemgt1.gt1 (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))
caucvgsrlemf.xfr 𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
Assertion
Ref Expression
caucvgsrlemcau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
Distinct variable groups:   𝑚,𝐹,𝑥   𝑦,𝐹,𝑥   𝑘,𝑚,𝑛,𝑥   𝜑,𝑘,𝑛,𝑥   𝑦,𝑘,𝑛   𝑛,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑦,𝑢,𝑚,𝑙)   𝐹(𝑢,𝑘,𝑛,𝑙)   𝐺(𝑥,𝑦,𝑢,𝑘,𝑚,𝑛,𝑙)

Proof of Theorem caucvgsrlemcau
StepHypRef Expression
1 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 ))))
2 caucvgsr.f . . . . . . . . . . 11 (𝜑𝐹:NR)
3 caucvgsrlemgt1.gt1 . . . . . . . . . . 11 (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))
4 caucvgsrlemf.xfr . . . . . . . . . . 11 𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
52, 1, 3, 4caucvgsrlemf 6933 . . . . . . . . . 10 (𝜑𝐺:NP)
65ad2antrr 465 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → 𝐺:NP)
7 simplr 490 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → 𝑛N)
86, 7ffvelrnd 5330 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → (𝐺𝑛) ∈ P)
95adantr 265 . . . . . . . . . 10 ((𝜑𝑛N) → 𝐺:NP)
109ffvelrnda 5329 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → (𝐺𝑘) ∈ P)
11 recnnpr 6703 . . . . . . . . . 10 (𝑛N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
127, 11syl 14 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
13 addclpr 6692 . . . . . . . . 9 (((𝐺𝑘) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
1410, 12, 13syl2anc 397 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
15 prsrlt 6928 . . . . . . . 8 (((𝐺𝑛) ∈ P ∧ ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P) → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ [⟨((𝐺𝑛) +P 1P), 1P⟩] ~R <R [⟨(((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R ))
168, 14, 15syl2anc 397 . . . . . . 7 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ [⟨((𝐺𝑛) +P 1P), 1P⟩] ~R <R [⟨(((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R ))
172, 1, 3, 4caucvgsrlemfv 6932 . . . . . . . . 9 ((𝜑𝑛N) → [⟨((𝐺𝑛) +P 1P), 1P⟩] ~R = (𝐹𝑛))
1817adantr 265 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → [⟨((𝐺𝑛) +P 1P), 1P⟩] ~R = (𝐹𝑛))
19 prsradd 6927 . . . . . . . . . 10 (((𝐺𝑘) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → [⟨(((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R = ([⟨((𝐺𝑘) +P 1P), 1P⟩] ~R +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
2010, 12, 19syl2anc 397 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → [⟨(((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R = ([⟨((𝐺𝑘) +P 1P), 1P⟩] ~R +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
212, 1, 3, 4caucvgsrlemfv 6932 . . . . . . . . . . 11 ((𝜑𝑘N) → [⟨((𝐺𝑘) +P 1P), 1P⟩] ~R = (𝐹𝑘))
2221adantlr 454 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → [⟨((𝐺𝑘) +P 1P), 1P⟩] ~R = (𝐹𝑘))
2322oveq1d 5554 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ([⟨((𝐺𝑘) +P 1P), 1P⟩] ~R +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
2420, 23eqtrd 2088 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → [⟨(((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R = ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
2518, 24breq12d 3804 . . . . . . 7 (((𝜑𝑛N) ∧ 𝑘N) → ([⟨((𝐺𝑛) +P 1P), 1P⟩] ~R <R [⟨(((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R ↔ (𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
2616, 25bitrd 181 . . . . . 6 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
27 addclpr 6692 . . . . . . . . 9 (((𝐺𝑛) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
288, 12, 27syl2anc 397 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P)
29 prsrlt 6928 . . . . . . . 8 (((𝐺𝑘) ∈ P ∧ ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∈ P) → ((𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ [⟨((𝐺𝑘) +P 1P), 1P⟩] ~R <R [⟨(((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R ))
3010, 28, 29syl2anc 397 . . . . . . 7 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ [⟨((𝐺𝑘) +P 1P), 1P⟩] ~R <R [⟨(((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R ))
31 prsradd 6927 . . . . . . . . . 10 (((𝐺𝑛) ∈ P ∧ ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P) → [⟨(((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R = ([⟨((𝐺𝑛) +P 1P), 1P⟩] ~R +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
328, 12, 31syl2anc 397 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → [⟨(((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R = ([⟨((𝐺𝑛) +P 1P), 1P⟩] ~R +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
3318oveq1d 5554 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ([⟨((𝐺𝑛) +P 1P), 1P⟩] ~R +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
3432, 33eqtrd 2088 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → [⟨(((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P 1P), 1P⟩] ~R = ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
3522, 34breq12d 3804 . . . . . . 7 (((𝜑𝑛N) ∧ 𝑘N) → ([⟨((𝐺𝑘) +P 1P), 1P⟩] ~R <R [⟨(((𝐺𝑛) +P ⟨{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 )))
3630, 35bitrd 181 . . . . . 6 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
3726, 36anbi12d 450 . . . . 5 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((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 ))))
3837imbi2d 223 . . . 4 (((𝜑𝑛N) ∧ 𝑘N) → ((𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))) ↔ (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 )))))
3938ralbidva 2339 . . 3 ((𝜑𝑛N) → (∀𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))) ↔ ∀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 )))))
4039ralbidva 2339 . 2 (𝜑 → (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))) ↔ ∀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 )))))
411, 40mpbird 160 1 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛)<P ((𝐺𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺𝑘)<P ((𝐺𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  {cab 2042  ∀wral 2323  ⟨cop 3405   class class class wbr 3791   ↦ cmpt 3845  ⟶wf 4925  ‘cfv 4929  ℩crio 5494  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427
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