Theorem axcaucvglemcau 8123

Description: Lemma for axcaucvg 8125. 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 8080	. . . . . . . . . 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 277	. . . . . . . . 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 4093	. . . . . . . . . . 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 5642	. . . . . . . . . . . . . 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 6038	. . . . . . . . . . . . 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 4101	. . . . . . . . . . . 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 4099	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 234	. . . . . . . . . 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 4092	. . . . . . . . . . . . 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 5642	. . . . . . . . . . . . . . 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 6030	. . . . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . . 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 6039	. . . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . . 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 6041	. . . . . . . . . . . . . . 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 4101	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 2531	. . . . . . . . . . 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 4092	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (𝑛 <ℝ 𝑘 ↔ 𝑎 <ℝ 𝑘))
24		fveq2 5642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
25		oveq1 6030	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑎 → (𝑛 · 𝑟) = (𝑎 · 𝑟))
26	25	eqeq1d 2239	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑎 → ((𝑛 · 𝑟) = 1 ↔ (𝑎 · 𝑟) = 1))
27	26	riotabidv 5978	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1))
28	27	oveq2d 6039	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
29	24, 28	breq12d 4102	. . . . . . . . . . . . . . . 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 6041	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑎) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
31	30	breq2d 4101	. . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 4093	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → (𝑎 <ℝ 𝑘 ↔ 𝑎 <ℝ 𝑏))
35		fveq2 5642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
36	35	oveq1d 6038	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)) = ((𝐹‘𝑏) + (℩𝑟 ∈ ℝ (𝑎 · 𝑟) = 1)))
37	36	breq2d 4101	. . . . . . . . . . . . . . . 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 4099	. . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 122	. . . . . . . . . . . 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 492	. . . . . . . . . . 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 8073	. . . . . . . . . . . . 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 2324	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑛, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑛, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
47	46	ad3antlr 493	. . . . . . . . . . 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 2914	. . . . . . . . . 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 8073	. . . . . . . . . . . 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 2324	. . . . . . . . . . 11 ⊢ (𝑘 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
51	50	ad2antlr 489	. . . . . . . . . 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 2914	. . . . . . . . 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 112	. . . . . . 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 8122	. . . . . . . 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 488	. . . . . . 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 8122	. . . . . . . . . . 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 477	. . . . . . . . . 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 276	. . . . . . . . 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 8115	. . . . . . . . . 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 493	. . . . . . . . 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 6041	. . . . . . . 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 8121	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:N⟶R)
66	65	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝐺:N⟶R)
67		simplr 529	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝑘 ∈ N)
68	66, 67	ffvelcdmd 5786	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑘) ∈ R)
69		recnnpr 7773	. . . . . . . . . . 11 ⊢ (𝑛 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
70		prsrcl 8009	. . . . . . . . . . 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 493	. . . . . . . . 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 8062	. . . . . . . . 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 411	. . . . . . . 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 2263	. . . . . . 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 4120	. . . . . 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 8064	. . . . . 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 122	. . . . 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 114	. . . . . . 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 6041	. . . . . . . 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 536	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → 𝑛 ∈ N)
82	66, 81	ffvelcdmd 5786	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ 𝑛 <N 𝑘) → (𝐺‘𝑛) ∈ R)
83		addresr 8062	. . . . . . . . 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 411	. . . . . . . 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 2263	. . . . . . 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 4120	. . . . . 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 8064	. . . . . 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 122	. . . . 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 306	. . . 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 115	. . 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 2604	. 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 2604	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 104   = wceq 1397   ∈ wcel 2201  {cab 2216  ∀wral 2509  ⟨cop 3673  ∩ cint 3929   class class class wbr 4089   ↦ cmpt 4151  ⟶wf 5324  ‘cfv 5328  ℩crio 5975  (class class class)co 6023  1_oc1o 6580  [cec 6705  Ncnpi 7497   <_N clti 7500   ~_Q ceq 7504  *_Qcrq 7509   <_Q cltq 7510  Pcnp 7516  1_Pc1p 7517   +_P cpp 7518   ~_R cer 7521  Rcnr 7522  0_Rc0r 7523   +_R cplr 7526   <_R cltr 7528  ℝcr 8036  1c1 8038   + caddc 8040   <_ℝ cltrr 8041   · cmul 8042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-eprel 4388  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-1o 6587  df-2o 6588  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-pli 7530  df-mi 7531  df-lti 7532  df-plpq 7569  df-mpq 7570  df-enq 7572  df-nqqs 7573  df-plqqs 7574  df-mqqs 7575  df-1nqqs 7576  df-rq 7577  df-ltnqqs 7578  df-enq0 7649  df-nq0 7650  df-0nq0 7651  df-plq0 7652  df-mq0 7653  df-inp 7691  df-i1p 7692  df-iplp 7693  df-imp 7694  df-iltp 7695  df-enr 7951  df-nr 7952  df-plr 7953  df-mr 7954  df-ltr 7955  df-0r 7956  df-1r 7957  df-m1r 7958  df-c 8043  df-0 8044  df-1 8045  df-r 8047  df-add 8048  df-mul 8049  df-lt 8050

This theorem is referenced by:  axcaucvglemres  8124