Theorem nninfdcinf 7512

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 5694	. . . . . 6 ⊢ (𝑓 = 𝑁 → (𝑓‘𝑥) = (𝑁‘𝑥))
2	1	eqeq1d 2247	. . . . 5 ⊢ (𝑓 = 𝑁 → ((𝑓‘𝑥) = 1o ↔ (𝑁‘𝑥) = 1o))
3	2	ralbidv 2550	. . . 4 ⊢ (𝑓 = 𝑁 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
4	3	dcbid 850	. . 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 2835	. . . . 5 ⊢ (𝜑 → ω ∈ V)
7		iswomnimap 7507	. . . . 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 7463	. . . . 5 ⊢ (𝑁 ∈ ℕ∞ → 𝑁:ω⟶2o)
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → 𝑁:ω⟶2o)
13		2onn 6794	. . . . . 6 ⊢ 2o ∈ ω
14	13	elexi 2834	. . . . 5 ⊢ 2o ∈ V
15		omex 4740	. . . . 5 ⊢ ω ∈ V
16	14, 15	elmap 6958	. . . 4 ⊢ (𝑁 ∈ (2o ↑𝑚 ω) ↔ 𝑁:ω⟶2o)
17	12, 16	sylibr 134	. . 3 ⊢ (𝜑 → 𝑁 ∈ (2o ↑𝑚 ω))
18	4, 9, 17	rspcdva 2934	. 2 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o)
19	12	ffnd 5534	. . . 4 ⊢ (𝜑 → 𝑁 Fn ω)
20		eqidd 2239	. . . 4 ⊢ (𝑥 = 𝑖 → 1o = 1o)
21		1onn 6793	. . . . 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 5813	. . 3 ⊢ (𝜑 → (𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝑁‘𝑥) = 1o))
25	24	dcbid 850	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ↦ cmpt 4192  ωcom 4737  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  1_oc1o 6680  2_oc2o 6681   ↑_𝑚 cmap 6922  ℕ_∞xnninf 7460  WOmnicwomni 7504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1o 6687  df-2o 6688  df-map 6924  df-nninf 7461  df-womni 7505

This theorem is used by:  nninfinfwlpo  7521