Definition df-bj-iomnn 38010

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 37957 but we did not use the present definition there since we wanted to have defined +∞ first.

See bj-iomnnom 38019 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 38009	. 2 class iω↪ℕ
2		vn	. . . 4 setvar 𝑛
3		com 7866	. . . 4 class ω
4		vr	. . . . . . . . . 10 setvar 𝑟
5	4	cv 1569	. . . . . . . . 9 class 𝑟
6	2	cv 1569	. . . . . . . . . . 11 class 𝑛
7	6	csuc 6363	. . . . . . . . . 10 class suc 𝑛
8		c1o 8452	. . . . . . . . . 10 class 1o
9	7, 8	cop 4593	. . . . . . . . 9 class ⟨suc 𝑛, 1o⟩
10		cltq 10871	. . . . . . . . 9 class <Q
11	5, 9, 10	wbr 5107	. . . . . . . 8 wff 𝑟 <Q ⟨suc 𝑛, 1o⟩
12		cnq 10865	. . . . . . . 8 class Q
13	11, 4, 12	crab 3414	. . . . . . 7 class {𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}
14		c1p 10873	. . . . . . 7 class 1P
15	13, 14	cop 4593	. . . . . 6 class ⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
16		cer 10877	. . . . . 6 class ~R
17	15, 16	cec 8698	. . . . 5 class [⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
18		c0r 10879	. . . . 5 class 0R
19	17, 18	cop 4593	. . . 4 class ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩
20	2, 3, 19	cmpt 5190	. . 3 class (𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩)
21		cpinfty 37979	. . . . 5 class +∞
22	3, 21	cop 4593	. . . 4 class ⟨ω, +∞⟩
23	22	csn 4587	. . 3 class {⟨ω, +∞⟩}
24	20, 23	cun 3900	. 2 class ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
25	1, 24	wceq 1570	1 wff iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})

This definition is used by:  bj-iomnnom  38019