Theorem nninfdcinf 7354

Description: The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ_∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)

Hypotheses

Ref	Expression
nninfdcinf.w	⊢ (𝜑 → ω ∈ WOmni)
nninfdcinf.n	⊢ (𝜑 → 𝑁 ∈ ℕ∞)

Assertion

Ref	Expression
nninfdcinf	⊢ (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))

Distinct variable groups:   𝑖,𝑁   𝜑,𝑖

Proof of Theorem nninfdcinf

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5631	. . . . . 6 ⊢ (𝑓 = 𝑁 → (𝑓‘𝑥) = (𝑁‘𝑥))
2	1	eqeq1d 2238	. . . . 5 ⊢ (𝑓 = 𝑁 → ((𝑓‘𝑥) = 1o ↔ (𝑁‘𝑥) = 1o))
3	2	ralbidv 2530	. . . 4 ⊢ (𝑓 = 𝑁 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
4	3	dcbid 843	. . 3 ⊢ (𝑓 = 𝑁 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
5		nninfdcinf.w	. . . 4 ⊢ (𝜑 → ω ∈ WOmni)
6	5	elexd 2813	. . . . 5 ⊢ (𝜑 → ω ∈ V)
7		iswomnimap 7349	. . . . 5 ⊢ (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o))
8	6, 7	syl 14	. . . 4 ⊢ (𝜑 → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o))
9	5, 8	mpbid 147	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o)
10		nninfdcinf.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ∞)
11		nninff 7305	. . . . 5 ⊢ (𝑁 ∈ ℕ∞ → 𝑁:ω⟶2o)
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → 𝑁:ω⟶2o)
13		2onn 6680	. . . . . 6 ⊢ 2o ∈ ω
14	13	elexi 2812	. . . . 5 ⊢ 2o ∈ V
15		omex 4686	. . . . 5 ⊢ ω ∈ V
16	14, 15	elmap 6837	. . . 4 ⊢ (𝑁 ∈ (2o ↑𝑚 ω) ↔ 𝑁:ω⟶2o)
17	12, 16	sylibr 134	. . 3 ⊢ (𝜑 → 𝑁 ∈ (2o ↑𝑚 ω))
18	4, 9, 17	rspcdva 2912	. 2 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)
19	12	ffnd 5477	. . . 4 ⊢ (𝜑 → 𝑁 Fn ω)
20		eqidd 2230	. . . 4 ⊢ (𝑥 = 𝑖 → 1o = 1o)
21		1onn 6679	. . . . 5 ⊢ 1o ∈ ω
22	21	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 1o ∈ ω)
23	21	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ ω)
24	19, 20, 22, 23	fnmptfvd 5744	. . 3 ⊢ (𝜑 → (𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
25	24	dcbid 843	. 2 ⊢ (𝜑 → (DECID 𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
26	18, 25	mpbird 167	1 ⊢ (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ↦ cmpt 4145  ωcom 4683  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010  1_oc1o 6566  2_oc2o 6567   ↑_𝑚 cmap 6808  ℕ_∞xnninf 7302  WOmnicwomni 7346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1o 6573  df-2o 6574  df-map 6810  df-nninf 7303  df-womni 7347

This theorem is referenced by:  nninfinfwlpo  7363