Definition df-bj-iomnn 37232

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 37179 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 37241 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 37231	. 2 class iω↪ℕ
2		vn	. . . 4 setvar 𝑛
3		com 7886	. . . 4 class ω
4		vr	. . . . . . . . . 10 setvar 𝑟
5	4	cv 1535	. . . . . . . . 9 class 𝑟
6	2	cv 1535	. . . . . . . . . . 11 class 𝑛
7	6	csuc 6387	. . . . . . . . . 10 class suc 𝑛
8		c1o 8497	. . . . . . . . . 10 class 1o
9	7, 8	cop 4636	. . . . . . . . 9 class ⟨suc 𝑛, 1o⟩
10		cltq 10895	. . . . . . . . 9 class <Q
11	5, 9, 10	wbr 5147	. . . . . . . 8 wff 𝑟 <Q ⟨suc 𝑛, 1o⟩
12		cnq 10889	. . . . . . . 8 class Q
13	11, 4, 12	crab 3432	. . . . . . 7 class {𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}
14		c1p 10897	. . . . . . 7 class 1P
15	13, 14	cop 4636	. . . . . 6 class ⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
16		cer 10901	. . . . . 6 class ~R
17	15, 16	cec 8741	. . . . 5 class [⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
18		c0r 10903	. . . . 5 class 0R
19	17, 18	cop 4636	. . . 4 class ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩
20	2, 3, 19	cmpt 5230	. . 3 class (𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩)
21		cpinfty 37201	. . . . 5 class +∞
22	3, 21	cop 4636	. . . 4 class ⟨ω, +∞⟩
23	22	csn 4630	. . 3 class {⟨ω, +∞⟩}
24	20, 23	cun 3960	. 2 class ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
25	1, 24	wceq 1536	1 wff iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

This definition is referenced by:  bj-iomnnom  37241