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Theorem axcaucvglemcau 7915
Description: Lemma for axcaucvg 7917. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
Hypotheses
Ref Expression
axcaucvg.n 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvg.f (𝜑𝐹:𝑁⟶ℝ)
axcaucvg.cau (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
axcaucvg.g 𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
Assertion
Ref Expression
axcaucvglemcau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Distinct variable groups:   𝑘,𝐹,𝑛,𝑧,𝑗   𝑘,𝑁,𝑛   𝑧,𝐺   𝑘,𝑙,𝑟,𝑢,𝑛   𝑗,𝑙,𝑢,𝑧   𝜑,𝑗,𝑘,𝑛   𝑦,𝑙,𝑢   𝑥,𝑦   𝑗,𝑛,𝑧,𝑘
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑢,𝑟,𝑙)   𝐹(𝑥,𝑦,𝑢,𝑟,𝑙)   𝐺(𝑥,𝑦,𝑢,𝑗,𝑘,𝑛,𝑟,𝑙)   𝑁(𝑥,𝑦,𝑧,𝑢,𝑗,𝑟,𝑙)

Proof of Theorem axcaucvglemcau
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrenn 7872 . . . . . . . . . 10 (𝑛 <N 𝑘 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
21adantl 277 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
3 breq2 4022 . . . . . . . . . . 11 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < 𝑏 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
4 fveq2 5530 . . . . . . . . . . . . . 14 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝐹𝑏) = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
54oveq1d 5906 . . . . . . . . . . . . 13 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
65breq2d 4030 . . . . . . . . . . . 12 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
74breq1d 4028 . . . . . . . . . . . 12 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
86, 7anbi12d 473 . . . . . . . . . . 11 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))) ↔ ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
93, 8imbi12d 234 . . . . . . . . . 10 (𝑏 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))) ↔ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))))
10 breq1 4021 . . . . . . . . . . . . 13 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑎 < 𝑏 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < 𝑏))
11 fveq2 5530 . . . . . . . . . . . . . . 15 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝐹𝑎) = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
12 oveq1 5898 . . . . . . . . . . . . . . . . . 18 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑎 · 𝑟) = (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟))
1312eqeq1d 2198 . . . . . . . . . . . . . . . . 17 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝑎 · 𝑟) = 1 ↔ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))
1413riotabidv 5849 . . . . . . . . . . . . . . . 16 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1) = (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))
1514oveq2d 5907 . . . . . . . . . . . . . . 15 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
1611, 15breq12d 4031 . . . . . . . . . . . . . 14 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
1711, 14oveq12d 5909 . . . . . . . . . . . . . . 15 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
1817breq2d 4030 . . . . . . . . . . . . . 14 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
1916, 18anbi12d 473 . . . . . . . . . . . . 13 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))) ↔ ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
2010, 19imbi12d 234 . . . . . . . . . . . 12 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))) ↔ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))))
2120ralbidv 2490 . . . . . . . . . . 11 (𝑎 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (∀𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))) ↔ ∀𝑏𝑁 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))))
22 axcaucvg.cau . . . . . . . . . . . . 13 (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
23 breq1 4021 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → (𝑛 < 𝑘𝑎 < 𝑘))
24 fveq2 5530 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → (𝐹𝑛) = (𝐹𝑎))
25 oveq1 5898 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
2625eqeq1d 2198 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
2726riotabidv 5849 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎 → (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
2827oveq2d 5907 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
2924, 28breq12d 4031 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3024, 27oveq12d 5909 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
3130breq2d 4030 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3229, 31anbi12d 473 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → (((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))) ↔ ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
3323, 32imbi12d 234 . . . . . . . . . . . . . 14 (𝑛 = 𝑎 → ((𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))) ↔ (𝑎 < 𝑘 → ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))))
34 breq2 4022 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → (𝑎 < 𝑘𝑎 < 𝑏))
35 fveq2 5530 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑏 → (𝐹𝑘) = (𝐹𝑏))
3635oveq1d 5906 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑏 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
3736breq2d 4030 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3835breq1d 4028 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3937, 38anbi12d 473 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → (((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))) ↔ ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
4034, 39imbi12d 234 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → ((𝑎 < 𝑘 → ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))) ↔ (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))))
4133, 40cbvral2v 2731 . . . . . . . . . . . . 13 (∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
4222, 41sylib 122 . . . . . . . . . . . 12 (𝜑 → ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
4342ad3antrrr 492 . . . . . . . . . . 11 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
44 pitonn 7865 . . . . . . . . . . . . 13 (𝑛N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
45 axcaucvg.n . . . . . . . . . . . . 13 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
4644, 45eleqtrrdi 2283 . . . . . . . . . . . 12 (𝑛N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
4746ad3antlr 493 . . . . . . . . . . 11 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
4821, 43, 47rspcdva 2861 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ∀𝑏𝑁 (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹𝑏) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
49 pitonn 7865 . . . . . . . . . . . 12 (𝑘N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
5049, 45eleqtrrdi 2283 . . . . . . . . . . 11 (𝑘N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
5150ad2antlr 489 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
529, 48, 51rspcdva 2861 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
532, 52mpd 13 . . . . . . . 8 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
5453simpld 112 . . . . . . 7 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
55 axcaucvg.f . . . . . . . . 9 (𝜑𝐹:𝑁⟶ℝ)
56 axcaucvg.g . . . . . . . . 9 𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
5745, 55, 22, 56axcaucvglemval 7914 . . . . . . . 8 ((𝜑𝑛N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑛), 0R⟩)
5857ad2antrr 488 . . . . . . 7 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑛), 0R⟩)
5945, 55, 22, 56axcaucvglemval 7914 . . . . . . . . . . 11 ((𝜑𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6059adantlr 477 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6160adantr 276 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
62 recriota 7907 . . . . . . . . . 10 (𝑛N → (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
6362ad3antlr 493 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
6461, 63oveq12d 5909 . . . . . . . 8 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = (⟨(𝐺𝑘), 0R⟩ + ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
6545, 55, 22, 56axcaucvglemf 7913 . . . . . . . . . . 11 (𝜑𝐺:NR)
6665ad3antrrr 492 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝐺:NR)
67 simplr 528 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑘N)
6866, 67ffvelcdmd 5668 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑘) ∈ R)
69 recnnpr 7565 . . . . . . . . . . 11 (𝑛N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
70 prsrcl 7801 . . . . . . . . . . 11 (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
7169, 70syl 14 . . . . . . . . . 10 (𝑛N → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
7271ad3antlr 493 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
73 addresr 7854 . . . . . . . . 9 (((𝐺𝑘) ∈ R ∧ [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR) → (⟨(𝐺𝑘), 0R⟩ + ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
7468, 72, 73syl2anc 411 . . . . . . . 8 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (⟨(𝐺𝑘), 0R⟩ + ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
7564, 74eqtrd 2222 . . . . . . 7 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = ⟨((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
7654, 58, 753brtr3d 4049 . . . . . 6 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨(𝐺𝑛), 0R⟩ < ⟨((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
77 ltresr 7856 . . . . . 6 (⟨(𝐺𝑛), 0R⟩ < ⟨((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩ ↔ (𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
7876, 77sylib 122 . . . . 5 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
7953simprd 114 . . . . . . 7 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) < ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
8058, 63oveq12d 5909 . . . . . . . 8 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = (⟨(𝐺𝑛), 0R⟩ + ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
81 simpllr 534 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑛N)
8266, 81ffvelcdmd 5668 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑛) ∈ R)
83 addresr 7854 . . . . . . . . 9 (((𝐺𝑛) ∈ R ∧ [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR) → (⟨(𝐺𝑛), 0R⟩ + ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
8482, 72, 83syl2anc 411 . . . . . . . 8 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (⟨(𝐺𝑛), 0R⟩ + ⟨[⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
8580, 84eqtrd 2222 . . . . . . 7 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = ⟨((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
8679, 61, 853brtr3d 4049 . . . . . 6 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨(𝐺𝑘), 0R⟩ < ⟨((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
87 ltresr 7856 . . . . . 6 (⟨(𝐺𝑘), 0R⟩ < ⟨((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩ ↔ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
8886, 87sylib 122 . . . . 5 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
8978, 88jca 306 . . . 4 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
9089ex 115 . . 3 (((𝜑𝑛N) ∧ 𝑘N) → (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
9190ralrimiva 2563 . 2 ((𝜑𝑛N) → ∀𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
9291ralrimiva 2563 1 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  {cab 2175  wral 2468  cop 3610   cint 3859   class class class wbr 4018  cmpt 4079  wf 5227  cfv 5231  crio 5846  (class class class)co 5891  1oc1o 6428  [cec 6551  Ncnpi 7289   <N clti 7292   ~Q ceq 7296  *Qcrq 7301   <Q cltq 7302  Pcnp 7308  1Pc1p 7309   +P cpp 7310   ~R cer 7313  Rcnr 7314  0Rc0r 7315   +R cplr 7318   <R cltr 7320  cr 7828  1c1 7830   + caddc 7832   < cltrr 7833   · cmul 7834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-pli 7322  df-mi 7323  df-lti 7324  df-plpq 7361  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-plqqs 7366  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-enq0 7441  df-nq0 7442  df-0nq0 7443  df-plq0 7444  df-mq0 7445  df-inp 7483  df-i1p 7484  df-iplp 7485  df-imp 7486  df-iltp 7487  df-enr 7743  df-nr 7744  df-plr 7745  df-mr 7746  df-ltr 7747  df-0r 7748  df-1r 7749  df-m1r 7750  df-c 7835  df-0 7836  df-1 7837  df-r 7839  df-add 7840  df-mul 7841  df-lt 7842
This theorem is referenced by:  axcaucvglemres  7916
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