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Theorem axcaucvglemcau 7627
Description: Lemma for axcaucvg 7629. 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 7584 . . . . . . . . . 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 273 . . . . . . . . 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 3897 . . . . . . . . . . 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 5373 . . . . . . . . . . . . . 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 5741 . . . . . . . . . . . . 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 3905 . . . . . . . . . . . 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 3903 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 233 . . . . . . . . . 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 3896 . . . . . . . . . . . . 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 5373 . . . . . . . . . . . . . . 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 5733 . . . . . . . . . . . . . . . . . 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 2121 . . . . . . . . . . . . . . . . 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 5684 . . . . . . . . . . . . . . . 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 5742 . . . . . . . . . . . . . . 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 3906 . . . . . . . . . . . . . 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 5744 . . . . . . . . . . . . . . 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 3905 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 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 233 . . . . . . . . . . . 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 2409 . . . . . . . . . . 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 3896 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → (𝑛 < 𝑘𝑎 < 𝑘))
24 fveq2 5373 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → (𝐹𝑛) = (𝐹𝑎))
25 oveq1 5733 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
2625eqeq1d 2121 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
2726riotabidv 5684 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎 → (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
2827oveq2d 5742 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
2924, 28breq12d 3906 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3024, 27oveq12d 5744 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
3130breq2d 3905 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3229, 31anbi12d 462 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 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 3897 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → (𝑎 < 𝑘𝑎 < 𝑏))
35 fveq2 5373 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑏 → (𝐹𝑘) = (𝐹𝑏))
3635oveq1d 5741 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑏 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
3736breq2d 3905 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3835breq1d 3903 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3937, 38anbi12d 462 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 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 2634 . . . . . . . . . . . . 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 121 . . . . . . . . . . . 12 (𝜑 → ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
4342ad3antrrr 481 . . . . . . . . . . 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 7577 . . . . . . . . . . . . 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, 45syl6eleqr 2206 . . . . . . . . . . . 12 (𝑛N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
4746ad3antlr 482 . . . . . . . . . . 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 2763 . . . . . . . . . 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 7577 . . . . . . . . . . . 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, 45syl6eleqr 2206 . . . . . . . . . . 11 (𝑘N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
5150ad2antlr 478 . . . . . . . . . 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 2763 . . . . . . . . 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 111 . . . . . . 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 7626 . . . . . . . 8 ((𝜑𝑛N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑛), 0R⟩)
5857ad2antrr 477 . . . . . . 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 7626 . . . . . . . . . . 11 ((𝜑𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6059adantlr 466 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6160adantr 272 . . . . . . . . 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 7619 . . . . . . . . . 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 482 . . . . . . . . 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 5744 . . . . . . . 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 7625 . . . . . . . . . . 11 (𝜑𝐺:NR)
6665ad3antrrr 481 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝐺:NR)
67 simplr 502 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑘N)
6866, 67ffvelrnd 5508 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑘) ∈ R)
69 recnnpr 7298 . . . . . . . . . . 11 (𝑛N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
70 prsrcl 7520 . . . . . . . . . . 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 482 . . . . . . . . 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 7566 . . . . . . . . 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 406 . . . . . . . 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 2145 . . . . . . 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 3922 . . . . . 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 7568 . . . . . 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 121 . . . . 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 113 . . . . . . 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 5744 . . . . . . . 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 506 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑛N)
8266, 81ffvelrnd 5508 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑛) ∈ R)
83 addresr 7566 . . . . . . . . 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 406 . . . . . . . 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 2145 . . . . . . 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 3922 . . . . . 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 7568 . . . . . 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 121 . . . . 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 302 . . . 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 114 . . 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 2477 . 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 2477 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 103   = wceq 1312  wcel 1461  {cab 2099  wral 2388  cop 3494   cint 3735   class class class wbr 3893  cmpt 3947  wf 5075  cfv 5079  crio 5681  (class class class)co 5726  1oc1o 6258  [cec 6379  Ncnpi 7022   <N clti 7025   ~Q ceq 7029  *Qcrq 7034   <Q cltq 7035  Pcnp 7041  1Pc1p 7042   +P cpp 7043   ~R cer 7046  Rcnr 7047  0Rc0r 7048   +R cplr 7051   <R cltr 7053  cr 7540  1c1 7542   + caddc 7544   < cltrr 7545   · cmul 7546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-i1p 7217  df-iplp 7218  df-imp 7219  df-iltp 7220  df-enr 7463  df-nr 7464  df-plr 7465  df-mr 7466  df-ltr 7467  df-0r 7468  df-1r 7469  df-m1r 7470  df-c 7547  df-0 7548  df-1 7549  df-r 7551  df-add 7552  df-mul 7553  df-lt 7554
This theorem is referenced by:  axcaucvglemres  7628
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