Definition df-bj-iomnn 37860

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 37807 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 37869 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 37859	. 2 class iω↪ℕ
2		vn	. . . 4 setvar 𝑛
3		com 7861	. . . 4 class ω
4		vr	. . . . . . . . . 10 setvar 𝑟
5	4	cv 1567	. . . . . . . . 9 class 𝑟
6	2	cv 1567	. . . . . . . . . . 11 class 𝑛
7	6	csuc 6362	. . . . . . . . . 10 class suc 𝑛
8		c1o 8445	. . . . . . . . . 10 class 1o
9	7, 8	cop 4594	. . . . . . . . 9 class ⟨suc 𝑛, 1o⟩
10		cltq 10842	. . . . . . . . 9 class <Q
11	5, 9, 10	wbr 5108	. . . . . . . 8 wff 𝑟 <Q ⟨suc 𝑛, 1o⟩
12		cnq 10836	. . . . . . . 8 class Q
13	11, 4, 12	crab 3414	. . . . . . 7 class {𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}
14		c1p 10844	. . . . . . 7 class 1P
15	13, 14	cop 4594	. . . . . 6 class ⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
16		cer 10848	. . . . . 6 class ~R
17	15, 16	cec 8691	. . . . 5 class [⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
18		c0r 10850	. . . . 5 class 0R
19	17, 18	cop 4594	. . . 4 class ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩
20	2, 3, 19	cmpt 5191	. . 3 class (𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩)
21		cpinfty 37829	. . . . 5 class +∞
22	3, 21	cop 4594	. . . 4 class ⟨ω, +∞⟩
23	22	csn 4588	. . 3 class {⟨ω, +∞⟩}
24	20, 23	cun 3902	. 2 class ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
25	1, 24	wceq 1568	1 wff iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

This definition is referenced by:  bj-iomnnom  37869