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Theorem axcaucvglemcau 7412
Description: Lemma for axcaucvg 7414. 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 7371 . . . . . . . . . 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 271 . . . . . . . . 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 3841 . . . . . . . . . . 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 5289 . . . . . . . . . . . . . 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 5649 . . . . . . . . . . . . 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 3849 . . . . . . . . . . . 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 3847 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 232 . . . . . . . . . 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 3840 . . . . . . . . . . . . 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 5289 . . . . . . . . . . . . . . 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 5641 . . . . . . . . . . . . . . . . . 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 2096 . . . . . . . . . . . . . . . . 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 5592 . . . . . . . . . . . . . . . 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 5650 . . . . . . . . . . . . . . 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 3850 . . . . . . . . . . . . . 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 5652 . . . . . . . . . . . . . . 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 3849 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 2380 . . . . . . . . . . 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 3840 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → (𝑛 < 𝑘𝑎 < 𝑘))
24 fveq2 5289 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → (𝐹𝑛) = (𝐹𝑎))
25 oveq1 5641 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
2625eqeq1d 2096 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
2726riotabidv 5592 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎 → (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
2827oveq2d 5650 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
2924, 28breq12d 3850 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3024, 27oveq12d 5652 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
3130breq2d 3849 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3229, 31anbi12d 457 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 3841 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → (𝑎 < 𝑘𝑎 < 𝑏))
35 fveq2 5289 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑏 → (𝐹𝑘) = (𝐹𝑏))
3635oveq1d 5649 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑏 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
3736breq2d 3849 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3835breq1d 3847 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
3937, 38anbi12d 457 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 2598 . . . . . . . . . . . . 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 120 . . . . . . . . . . . 12 (𝜑 → ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
4342ad3antrrr 476 . . . . . . . . . . 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 7364 . . . . . . . . . . . . 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 2181 . . . . . . . . . . . 12 (𝑛N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
4746ad3antlr 477 . . . . . . . . . . 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 2727 . . . . . . . . . 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 7364 . . . . . . . . . . . 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 2181 . . . . . . . . . . 11 (𝑘N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
5150ad2antlr 473 . . . . . . . . . 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 2727 . . . . . . . . 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 110 . . . . . . 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 7411 . . . . . . . 8 ((𝜑𝑛N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑛), 0R⟩)
5857ad2antrr 472 . . . . . . 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 7411 . . . . . . . . . . 11 ((𝜑𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6059adantlr 461 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6160adantr 270 . . . . . . . . 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 7404 . . . . . . . . . 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 477 . . . . . . . . 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 5652 . . . . . . . 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 7410 . . . . . . . . . . 11 (𝜑𝐺:NR)
6665ad3antrrr 476 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝐺:NR)
67 simplr 497 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑘N)
6866, 67ffvelrnd 5419 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑘) ∈ R)
69 recnnpr 7086 . . . . . . . . . . 11 (𝑛N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
70 prsrcl 7308 . . . . . . . . . . 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 477 . . . . . . . . 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 7353 . . . . . . . . 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 403 . . . . . . . 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 2120 . . . . . . 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 3866 . . . . . 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 7355 . . . . . 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 120 . . . . 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 112 . . . . . . 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 5652 . . . . . . . 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 501 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑛N)
8266, 81ffvelrnd 5419 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑛) ∈ R)
83 addresr 7353 . . . . . . . . 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 403 . . . . . . . 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 2120 . . . . . . 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 3866 . . . . . 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 7355 . . . . . 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 120 . . . . 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 300 . . . 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 113 . . 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 2446 . 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 2446 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 102   = wceq 1289  wcel 1438  {cab 2074  wral 2359  cop 3444   cint 3683   class class class wbr 3837  cmpt 3891  wf 4998  cfv 5002  crio 5589  (class class class)co 5634  1𝑜c1o 6156  [cec 6270  Ncnpi 6810   <N clti 6813   ~Q ceq 6817  *Qcrq 6822   <Q cltq 6823  Pcnp 6829  1Pc1p 6830   +P cpp 6831   ~R cer 6834  Rcnr 6835  0Rc0r 6836   +R cplr 6839   <R cltr 6841  cr 7328  1c1 7330   + caddc 7332   < cltrr 7333   · cmul 7334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-i1p 7005  df-iplp 7006  df-imp 7007  df-iltp 7008  df-enr 7251  df-nr 7252  df-plr 7253  df-mr 7254  df-ltr 7255  df-0r 7256  df-1r 7257  df-m1r 7258  df-c 7335  df-0 7336  df-1 7337  df-r 7339  df-add 7340  df-mul 7341  df-lt 7342
This theorem is referenced by:  axcaucvglemres  7413
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