Theorem nninfdcinf 7306

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 5602	. . . . . 6 ⊢ (𝑓 = 𝑁 → (𝑓‘𝑥) = (𝑁‘𝑥))
2	1	eqeq1d 2218	. . . . 5 ⊢ (𝑓 = 𝑁 → ((𝑓‘𝑥) = 1o ↔ (𝑁‘𝑥) = 1o))
3	2	ralbidv 2510	. . . 4 ⊢ (𝑓 = 𝑁 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
4	3	dcbid 842	. . 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 2793	. . . . 5 ⊢ (𝜑 → ω ∈ V)
7		iswomnimap 7301	. . . . 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 7257	. . . . 5 ⊢ (𝑁 ∈ ℕ∞ → 𝑁:ω⟶2o)
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → 𝑁:ω⟶2o)
13		2onn 6637	. . . . . 6 ⊢ 2o ∈ ω
14	13	elexi 2792	. . . . 5 ⊢ 2o ∈ V
15		omex 4662	. . . . 5 ⊢ ω ∈ V
16	14, 15	elmap 6794	. . . 4 ⊢ (𝑁 ∈ (2o ↑𝑚 ω) ↔ 𝑁:ω⟶2o)
17	12, 16	sylibr 134	. . 3 ⊢ (𝜑 → 𝑁 ∈ (2o ↑𝑚 ω))
18	4, 9, 17	rspcdva 2892	. 2 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)
19	12	ffnd 5450	. . . 4 ⊢ (𝜑 → 𝑁 Fn ω)
20		eqidd 2210	. . . 4 ⊢ (𝑥 = 𝑖 → 1o = 1o)
21		1onn 6636	. . . . 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 5712	. . 3 ⊢ (𝜑 → (𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
25	24	dcbid 842	. 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 838   = wceq 1375   ∈ wcel 2180  ∀wral 2488  Vcvv 2779   ↦ cmpt 4124  ωcom 4659  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974  1_oc1o 6525  2_oc2o 6526   ↑_𝑚 cmap 6765  ℕ_∞xnninf 7254  WOmnicwomni 7298

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1o 6532  df-2o 6533  df-map 6767  df-nninf 7255  df-womni 7299

This theorem is referenced by:  nninfinfwlpo  7315