Theorem nninfdcinf 7475

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 5674	. . . . . 6 ⊢ (𝑓 = 𝑁 → (𝑓‘𝑥) = (𝑁‘𝑥))
2	1	eqeq1d 2243	. . . . 5 ⊢ (𝑓 = 𝑁 → ((𝑓‘𝑥) = 1o ↔ (𝑁‘𝑥) = 1o))
3	2	ralbidv 2544	. . . 4 ⊢ (𝑓 = 𝑁 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
4	3	dcbid 846	. . 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 2829	. . . . 5 ⊢ (𝜑 → ω ∈ V)
7		iswomnimap 7470	. . . . 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 7426	. . . . 5 ⊢ (𝑁 ∈ ℕ∞ → 𝑁:ω⟶2o)
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → 𝑁:ω⟶2o)
13		2onn 6767	. . . . . 6 ⊢ 2o ∈ ω
14	13	elexi 2828	. . . . 5 ⊢ 2o ∈ V
15		omex 4720	. . . . 5 ⊢ ω ∈ V
16	14, 15	elmap 6924	. . . 4 ⊢ (𝑁 ∈ (2o ↑𝑚 ω) ↔ 𝑁:ω⟶2o)
17	12, 16	sylibr 134	. . 3 ⊢ (𝜑 → 𝑁 ∈ (2o ↑𝑚 ω))
18	4, 9, 17	rspcdva 2928	. 2 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)
19	12	ffnd 5514	. . . 4 ⊢ (𝜑 → 𝑁 Fn ω)
20		eqidd 2235	. . . 4 ⊢ (𝑥 = 𝑖 → 1o = 1o)
21		1onn 6766	. . . . 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 5787	. . 3 ⊢ (𝜑 → (𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
25	24	dcbid 846	. 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 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ↦ cmpt 4176  ωcom 4717  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1_oc1o 6653  2_oc2o 6654   ↑_𝑚 cmap 6895  ℕ_∞xnninf 7423  WOmnicwomni 7467

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1o 6660  df-2o 6661  df-map 6897  df-nninf 7424  df-womni 7468

This theorem is referenced by:  nninfinfwlpo  7484