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 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvg.f	⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
axcaucvg.cau	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
axcaucvg.g	⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))

Assertion

Ref	Expression
axcaucvglemcau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <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.

Step	Hyp	Ref	Expression
1		ltrenn 7584	. . . . . . . . . 10 ⊢ (𝑛 <N 𝑘 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
2	1	adantl 273	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
3		breq2 3897	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ 𝑏 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
4		fveq2 5373	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝐹‘𝑏) = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
5	4	oveq1d 5741	. . . . . . . . . . . . 13 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
6	5	breq2d 3905	. . . . . . . . . . . 12 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
7	4	breq1d 3903	. . . . . . . . . . . 12 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
8	6, 7	anbi12d 462	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))) ↔ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
9	3, 8	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑏 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))) ↔ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))))
10		breq1 3896	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑎 <ℝ 𝑏 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ 𝑏))
11		fveq2 5373	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝐹‘𝑎) = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
12		oveq1 5733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑎 · 𝑟) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟))
13	12	eqeq1d 2121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝑎 · 𝑟) = 1 ↔ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))
14	13	riotabidv 5684	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1) = (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))
15	14	oveq2d 5742	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
16	11, 15	breq12d 3906	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
17	11, 14	oveq12d 5744	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
18	17	breq2d 3905	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
19	16, 18	anbi12d 462	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))) ↔ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
20	10, 19	imbi12d 233	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝑎 <ℝ 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))) ↔ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))))
21	20	ralbidv 2409	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (∀𝑏 ∈ 𝑁 (𝑎 <ℝ 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))) ↔ ∀𝑏 ∈ 𝑁 (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))))
22		axcaucvg.cau	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
23		breq1 3896	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (𝑛 <ℝ 𝑘 ↔ 𝑎 <ℝ 𝑘))
24		fveq2 5373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
25		oveq1 5733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
26	25	eqeq1d 2121	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
27	26	riotabidv 5684	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
28	27	oveq2d 5742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
29	24, 28	breq12d 3906	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹‘𝑎) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
30	24, 27	oveq12d 5744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
31	30	breq2d 3905	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹‘𝑘) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
32	29, 31	anbi12d 462	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))) ↔ ((𝐹‘𝑎) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
33	23, 32	imbi12d 233	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑎 → ((𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))) ↔ (𝑎 <ℝ 𝑘 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))))
34		breq2 3897	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → (𝑎 <ℝ 𝑘 ↔ 𝑎 <ℝ 𝑏))
35		fveq2 5373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
36	35	oveq1d 5741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
37	36	breq2d 3905	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
38	35	breq1d 3903	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ↔ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))
39	37, 38	anbi12d 462	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → (((𝐹‘𝑎) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))) ↔ ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
40	34, 39	imbi12d 233	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑏 → ((𝑎 <ℝ 𝑘 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))) ↔ (𝑎 <ℝ 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))))))
41	33, 40	cbvral2v 2634	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))) ↔ ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎 <ℝ 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
42	22, 41	sylib 121	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎 <ℝ 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
43	42	ad3antrrr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ∀𝑎 ∈ 𝑁 ∀𝑏 ∈ 𝑁 (𝑎 <ℝ 𝑏 → ((𝐹‘𝑎) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))))
44		pitonn 7577	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
45		axcaucvg.n	. . . . . . . . . . . . 13 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
46	44, 45	syl6eleqr 2206	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
47	46	ad3antlr 482	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
48	21, 43, 47	rspcdva 2763	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ∀𝑏 ∈ 𝑁 (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ 𝑏 → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘𝑏) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
49		pitonn 7577	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
50	49, 45	syl6eleqr 2206	. . . . . . . . . . 11 ⊢ (𝑘 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
51	50	ad2antlr 478	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
52	9, 48, 51	rspcdva 2763	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))))
53	2, 52	mpd 13	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) ∧ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1))))
54	53	simpld 111	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
55		axcaucvg.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
56		axcaucvg.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
57	45, 55, 22, 56	axcaucvglemval 7626	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝑛), 0R⟩)
58	57	ad2antrr 477	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝑛), 0R⟩)
59	45, 55, 22, 56	axcaucvglemval 7626	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝑘), 0R⟩)
60	59	adantlr 466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝑘), 0R⟩)
61	60	adantr 272	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝑘), 0R⟩)
62		recriota 7619	. . . . . . . . . 10 ⊢ (𝑛 ∈ N → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
63	62	ad3antlr 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
64	61, 63	oveq12d 5744	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = (⟨(𝐺‘𝑘), 0R⟩ + ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
65	45, 55, 22, 56	axcaucvglemf 7625	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:N⟶R)
66	65	ad3antrrr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝐺:N⟶R)
67		simplr 502	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝑘 ∈ N)
68	66, 67	ffvelrnd 5508	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑘) ∈ R)
69		recnnpr 7298	. . . . . . . . . . 11 ⊢ (𝑛 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
70		prsrcl 7520	. . . . . . . . . . 11 ⊢ (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
71	69, 70	syl 14	. . . . . . . . . 10 ⊢ (𝑛 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
72	71	ad3antlr 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
73		addresr 7566	. . . . . . . . 9 ⊢ (((𝐺‘𝑘) ∈ R ∧ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R) → (⟨(𝐺‘𝑘), 0R⟩ + ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
74	68, 72, 73	syl2anc 406	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (⟨(𝐺‘𝑘), 0R⟩ + ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
75	64, 74	eqtrd 2145	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = ⟨((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
76	54, 58, 75	3brtr3d 3922	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ⟨(𝐺‘𝑛), 0R⟩ <ℝ ⟨((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
77		ltresr 7568	. . . . . 6 ⊢ (⟨(𝐺‘𝑛), 0R⟩ <ℝ ⟨((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩ ↔ (𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
78	76, 77	sylib 121	. . . . 5 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
79	53	simprd 113	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) <ℝ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)))
80	58, 63	oveq12d 5744	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = (⟨(𝐺‘𝑛), 0R⟩ + ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
81		simpllr 506	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝑛 ∈ N)
82	66, 81	ffvelrnd 5508	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑛) ∈ R)
83		addresr 7566	. . . . . . . . 9 ⊢ (((𝐺‘𝑛) ∈ R ∧ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R) → (⟨(𝐺‘𝑛), 0R⟩ + ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
84	82, 72, 83	syl2anc 406	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (⟨(𝐺‘𝑛), 0R⟩ + ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
85	80, 84	eqtrd 2145	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) + (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)) = ⟨((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
86	79, 61, 85	3brtr3d 3922	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ⟨(𝐺‘𝑘), 0R⟩ <ℝ ⟨((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩)
87		ltresr 7568	. . . . . 6 ⊢ (⟨(𝐺‘𝑘), 0R⟩ <ℝ ⟨((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ), 0R⟩ ↔ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
88	86, 87	sylib 121	. . . . 5 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
89	78, 88	jca 302	. . . 4 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
90	89	ex 114	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
91	90	ralrimiva 2477	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ N) → ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
92	91	ralrimiva 2477	1 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))

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  1_oc1o 6258  [cec 6379  Ncnpi 7022   <_N clti 7025   ~_Q ceq 7029  *_Qcrq 7034   <_Q cltq 7035  Pcnp 7041  1_Pc1p 7042   +_P cpp 7043   ~_R cer 7046  Rcnr 7047  0_Rc0r 7048   +_R cplr 7051   <_R cltr 7053  ℝcr 7540  1c1 7542   + caddc 7544   <_ℝ 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