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Theorem axcaucvglemcau 6972
 Description: Lemma for axcaucvg 6974. 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 6931 . . . . . . . . . 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 262 . . . . . . . . 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 pitonn 6924 . . . . . . . . . . . 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)})
4 axcaucvg.n . . . . . . . . . . . 12 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
53, 4syl6eleqr 2131 . . . . . . . . . . 11 (𝑘N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
65ad2antlr 458 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
7 pitonn 6924 . . . . . . . . . . . . 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)})
87, 4syl6eleqr 2131 . . . . . . . . . . . 12 (𝑛N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
98ad3antlr 462 . . . . . . . . . . 11 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
10 axcaucvg.cau . . . . . . . . . . . . 13 (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
11 breq1 3767 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → (𝑛 < 𝑘𝑎 < 𝑘))
12 fveq2 5178 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → (𝐹𝑛) = (𝐹𝑎))
13 oveq1 5519 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
1413eqeq1d 2048 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
1514riotabidv 5470 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎 → (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
1615oveq2d 5528 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
1712, 16breq12d 3777 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
1812, 15oveq12d 5530 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
1918breq2d 3776 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
2017, 19anbi12d 442 . . . . . . . . . . . . . . 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)))))
2111, 20imbi12d 223 . . . . . . . . . . . . . 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))))))
22 breq2 3768 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → (𝑎 < 𝑘𝑎 < 𝑏))
23 fveq2 5178 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑏 → (𝐹𝑘) = (𝐹𝑏))
2423oveq1d 5527 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑏 → ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
2524breq2d 3776 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑎) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
2623breq1d 3774 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑘) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
2725, 26anbi12d 442 . . . . . . . . . . . . . . 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)))))
2822, 27imbi12d 223 . . . . . . . . . . . . . 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))))))
2921, 28cbvral2v 2541 . . . . . . . . . . . . 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)))))
3010, 29sylib 127 . . . . . . . . . . . 12 (𝜑 → ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
3130ad3antrrr 461 . . . . . . . . . . 11 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → ∀𝑎𝑁𝑏𝑁 (𝑎 < 𝑏 → ((𝐹𝑎) < ((𝐹𝑏) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹𝑏) < ((𝐹𝑎) + (𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
32 breq1 3767 . . . . . . . . . . . . . 14 (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))
33 fveq2 5178 . . . . . . . . . . . . . . . 16 (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⟩))
34 oveq1 5519 . . . . . . . . . . . . . . . . . . 19 (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))
3534eqeq1d 2048 . . . . . . . . . . . . . . . . . 18 (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))
3635riotabidv 5470 . . . . . . . . . . . . . . . . 17 (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))
3736oveq2d 5528 . . . . . . . . . . . . . . . 16 (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)))
3833, 37breq12d 3777 . . . . . . . . . . . . . . 15 (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))))
3933, 36oveq12d 5530 . . . . . . . . . . . . . . . 16 (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)))
4039breq2d 3776 . . . . . . . . . . . . . . 15 (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))))
4138, 40anbi12d 442 . . . . . . . . . . . . . 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)) ∧ (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)))))
4232, 41imbi12d 223 . . . . . . . . . . . . 13 (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))))))
4342ralbidv 2326 . . . . . . . . . . . 12 (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))))))
4443rspcva 2654 . . . . . . . . . . 11 ((⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁 ∧ ∀𝑎𝑁𝑏𝑁 (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)))))
459, 31, 44syl2anc 391 . . . . . . . . . 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)))))
46 breq2 3768 . . . . . . . . . . . 12 (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⟩))
47 fveq2 5178 . . . . . . . . . . . . . . 15 (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⟩))
4847oveq1d 5527 . . . . . . . . . . . . . 14 (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)))
4948breq2d 3776 . . . . . . . . . . . . 13 (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))))
5047breq1d 3774 . . . . . . . . . . . . 13 (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))))
5149, 50anbi12d 442 . . . . . . . . . . . 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)) ∧ (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)))))
5246, 51imbi12d 223 . . . . . . . . . . 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 → ((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))))))
5352rspcva 2654 . . . . . . . . . 10 ((⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁 ∧ ∀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))))) → (⟨[⟨(⟨{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)))))
546, 45, 53syl2anc 391 . . . . . . . . 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)))))
552, 54mpd 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))))
5655simpld 105 . . . . . . 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)))
57 axcaucvg.f . . . . . . . . 9 (𝜑𝐹:𝑁⟶ℝ)
58 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⟩))
594, 57, 10, 58axcaucvglemval 6971 . . . . . . . 8 ((𝜑𝑛N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑛, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑛), 0R⟩)
6059ad2antrr 457 . . . . . . 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⟩)
614, 57, 10, 58axcaucvglemval 6971 . . . . . . . . . . 11 ((𝜑𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6261adantlr 446 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝑘), 0R⟩)
6362adantr 261 . . . . . . . . 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⟩)
64 recriota 6964 . . . . . . . . . 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⟩)
6564ad3antlr 462 . . . . . . . . 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⟩)
6663, 65oveq12d 5530 . . . . . . . 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⟩))
674, 57, 10, 58axcaucvglemf 6970 . . . . . . . . . . 11 (𝜑𝐺:NR)
6867ad3antrrr 461 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝐺:NR)
69 simplr 482 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑘N)
7068, 69ffvelrnd 5303 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑘) ∈ R)
71 recnnpr 6646 . . . . . . . . . . 11 (𝑛N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
72 prsrcl 6868 . . . . . . . . . . 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)
7371, 72syl 14 . . . . . . . . . 10 (𝑛N → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
7473ad3antlr 462 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
75 addresr 6913 . . . . . . . . 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⟩)
7670, 74, 75syl2anc 391 . . . . . . . 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⟩)
7766, 76eqtrd 2072 . . . . . . 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⟩)
7856, 60, 773brtr3d 3793 . . . . . 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⟩)
79 ltresr 6915 . . . . . 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 ))
8078, 79sylib 127 . . . . 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 ))
8155simprd 107 . . . . . . 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)))
8260, 65oveq12d 5530 . . . . . . . 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⟩))
83 simpllr 486 . . . . . . . . . 10 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → 𝑛N)
8468, 83ffvelrnd 5303 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ 𝑛 <N 𝑘) → (𝐺𝑛) ∈ R)
85 addresr 6913 . . . . . . . . 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⟩)
8684, 74, 85syl2anc 391 . . . . . . . 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⟩)
8782, 86eqtrd 2072 . . . . . . 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⟩)
8881, 63, 873brtr3d 3793 . . . . . 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⟩)
89 ltresr 6915 . . . . . 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 ))
9088, 89sylib 127 . . . . 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 ))
9180, 90jca 290 . . . 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 )))
9291ex 108 . . 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 ))))
9392ralrimiva 2392 . 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 ))))
9493ralrimiva 2392 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 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ⟨cop 3378  ∩ cint 3615   class class class wbr 3764   ↦ cmpt 3818  ⟶wf 4898  ‘cfv 4902  ℩crio 5467  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6370
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