Definition df-bj-iomnn 37754

Description: Definition of the canonical bijection from (ω ∪ {ω}) onto (ℕ₀ ∪ {+∞}).

To understand this definition, recall that set.mm constructs reals as couples whose first component is a prereal and second component is the zero prereal (in order that one have ℝ ⊆ ℂ), that prereals are equivalence classes of couples of positive reals, the latter are Dedekind cuts of positive rationals, which are equivalence classes of positive ordinals. In partiular, we take the successor ordinal at the beginning and subtract 1 at the end since the intermediate systems contain only (strictly) positive numbers.

Note the similarity with df-bj-fractemp 37701 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 37763 for its value at +∞.

TODO:

Prove ⊢ (i_ω↪ℕ‘∅) = 0.

Define ⊢ ℕ₀ = (i_ω↪ℕ “ ω) and ⊢ ℕ = (ℕ₀ ∖ {0}).

Prove ⊢ i_ω↪ℕ:(ω ∪ {ω})–1-1-onto→(ℕ₀ ∪ {+∞}) and ⊢ (i_ω↪ℕ ↾ ω):ω–1-1-onto→ℕ₀.

Prove that these bijections are respectively an isomorphism of ordered "extended rigs" and of ordered rigs.

Prove ⊢ (i_ω↪ℕ ↾ ω) = rec((𝑥 ∈ ℝ ↦ (𝑥 + 1)), 0).

(Contributed by BJ, 18-Feb-2023.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)

Assertion

Ref	Expression
df-bj-iomnn	⊢ iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

Distinct variable group:   𝑛,𝑟

Detailed syntax breakdown of Definition df-bj-iomnn

Step	Hyp	Ref	Expression
1		ciomnn 37753	. 2 class iω↪ℕ
2		vn	. . . 4 setvar 𝑛
3		com 7850	. . . 4 class ω
4		vr	. . . . . . . . . 10 setvar 𝑟
5	4	cv 1562	. . . . . . . . 9 class 𝑟
6	2	cv 1562	. . . . . . . . . . 11 class 𝑛
7	6	csuc 6352	. . . . . . . . . 10 class suc 𝑛
8		c1o 8434	. . . . . . . . . 10 class 1o
9	7, 8	cop 4591	. . . . . . . . 9 class ⟨suc 𝑛, 1o⟩
10		cltq 10831	. . . . . . . . 9 class <Q
11	5, 9, 10	wbr 5105	. . . . . . . 8 wff 𝑟 <Q ⟨suc 𝑛, 1o⟩
12		cnq 10825	. . . . . . . 8 class Q
13	11, 4, 12	crab 3417	. . . . . . 7 class {𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}
14		c1p 10833	. . . . . . 7 class 1P
15	13, 14	cop 4591	. . . . . 6 class ⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
16		cer 10837	. . . . . 6 class ~R
17	15, 16	cec 8680	. . . . 5 class [⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
18		c0r 10839	. . . . 5 class 0R
19	17, 18	cop 4591	. . . 4 class ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩
20	2, 3, 19	cmpt 5186	. . 3 class (𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩)
21		cpinfty 37723	. . . . 5 class +∞
22	3, 21	cop 4591	. . . 4 class ⟨ω, +∞⟩
23	22	csn 4585	. . 3 class {⟨ω, +∞⟩}
24	20, 23	cun 3905	. 2 class ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
25	1, 24	wceq 1563	1 wff iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

This definition is referenced by:  bj-iomnnom  37763