Theorem axcaucvglemcau 7839

Description: Lemma for axcaucvg 7841. 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 7796	. . . . . . . . . 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 275	. . . . . . . . 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 3986	. . . . . . . . . . 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 5486	. . . . . . . . . . . . . 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 5857	. . . . . . . . . . . . 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 3994	. . . . . . . . . . . 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 3992	. . . . . . . . . . . 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 465	. . . . . . . . . . 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 3985	. . . . . . . . . . . . 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 5486	. . . . . . . . . . . . . . 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 5849	. . . . . . . . . . . . . . . . . 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 2174	. . . . . . . . . . . . . . . . 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 5800	. . . . . . . . . . . . . . . 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 5858	. . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . 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 5860	. . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . . 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 465	. . . . . . . . . . . . 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 2466	. . . . . . . . . . 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 3985	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (𝑛 <ℝ 𝑘 ↔ 𝑎 <ℝ 𝑘))
24		fveq2 5486	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
25		oveq1 5849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
26	25	eqeq1d 2174	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
27	26	riotabidv 5800	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
28	27	oveq2d 5858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
29	24, 28	breq12d 3995	. . . . . . . . . . . . . . . 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 5860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
31	30	breq2d 3994	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 3986	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → (𝑎 <ℝ 𝑘 ↔ 𝑎 <ℝ 𝑏))
35		fveq2 5486	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
36	35	oveq1d 5857	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
37	36	breq2d 3994	. . . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 2705	. . . . . . . . . . . . 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 484	. . . . . . . . . . 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 7789	. . . . . . . . . . . . 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	eleqtrrdi 2260	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
47	46	ad3antlr 485	. . . . . . . . . . 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 2835	. . . . . . . . . 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 7789	. . . . . . . . . . . 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	eleqtrrdi 2260	. . . . . . . . . . 11 ⊢ (𝑘 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
51	50	ad2antlr 481	. . . . . . . . . 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 2835	. . . . . . . . 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 7838	. . . . . . . 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 480	. . . . . . 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 7838	. . . . . . . . . . 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 469	. . . . . . . . . 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 274	. . . . . . . . 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 7831	. . . . . . . . . 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 485	. . . . . . . . 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 5860	. . . . . . . 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 7837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:N⟶R)
66	65	ad3antrrr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝐺:N⟶R)
67		simplr 520	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝑘 ∈ N)
68	66, 67	ffvelrnd 5621	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑘) ∈ R)
69		recnnpr 7489	. . . . . . . . . . 11 ⊢ (𝑛 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
70		prsrcl 7725	. . . . . . . . . . 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 485	. . . . . . . . 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 7778	. . . . . . . . 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 409	. . . . . . . 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 2198	. . . . . . 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 4013	. . . . . 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 7780	. . . . . 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 5860	. . . . . . . 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 524	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝑛 ∈ N)
82	66, 81	ffvelrnd 5621	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑛) ∈ R)
83		addresr 7778	. . . . . . . . 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 409	. . . . . . . 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 2198	. . . . . . 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 4013	. . . . . 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 7780	. . . . . 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 304	. . . 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 2539	. 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 2539	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 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ⟨cop 3579  ∩ cint 3824   class class class wbr 3982   ↦ cmpt 4043  ⟶wf 5184  ‘cfv 5188  ℩crio 5797  (class class class)co 5842  1_oc1o 6377  [cec 6499  Ncnpi 7213   <_N clti 7216   ~_Q ceq 7220  *_Qcrq 7225   <_Q cltq 7226  Pcnp 7232  1_Pc1p 7233   +_P cpp 7234   ~_R cer 7237  Rcnr 7238  0_Rc0r 7239   +_R cplr 7242   <_R cltr 7244  ℝcr 7752  1c1 7754   + caddc 7756   <_ℝ cltrr 7757   · cmul 7758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-imp 7410  df-iltp 7411  df-enr 7667  df-nr 7668  df-plr 7669  df-mr 7670  df-ltr 7671  df-0r 7672  df-1r 7673  df-m1r 7674  df-c 7759  df-0 7760  df-1 7761  df-r 7763  df-add 7764  df-mul 7765  df-lt 7766

This theorem is referenced by:  axcaucvglemres  7840