Definition df-bj-iomnn 37231

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 37178 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 37240 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 37230	. 2 class iω↪ℕ
2		vn	. . . 4 setvar 𝑛
3		com 7822	. . . 4 class ω
4		vr	. . . . . . . . . 10 setvar 𝑟
5	4	cv 1539	. . . . . . . . 9 class 𝑟
6	2	cv 1539	. . . . . . . . . . 11 class 𝑛
7	6	csuc 6322	. . . . . . . . . 10 class suc 𝑛
8		c1o 8404	. . . . . . . . . 10 class 1o
9	7, 8	cop 4591	. . . . . . . . 9 class ⟨suc 𝑛, 1o⟩
10		cltq 10787	. . . . . . . . 9 class <Q
11	5, 9, 10	wbr 5102	. . . . . . . 8 wff 𝑟 <Q ⟨suc 𝑛, 1o⟩
12		cnq 10781	. . . . . . . 8 class Q
13	11, 4, 12	crab 3402	. . . . . . 7 class {𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}
14		c1p 10789	. . . . . . 7 class 1P
15	13, 14	cop 4591	. . . . . 6 class ⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
16		cer 10793	. . . . . 6 class ~R
17	15, 16	cec 8646	. . . . 5 class [⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
18		c0r 10795	. . . . 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 5183	. . 3 class (𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩)
21		cpinfty 37200	. . . . 5 class +∞
22	3, 21	cop 4591	. . . 4 class ⟨ω, +∞⟩
23	22	csn 4585	. . 3 class {⟨ω, +∞⟩}
24	20, 23	cun 3909	. 2 class ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
25	1, 24	wceq 1540	1 wff iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

This definition is referenced by:  bj-iomnnom  37240