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Theorem nninfdcinf 7162
Description: The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
Hypotheses
Ref Expression
nninfdcinf.w (𝜑 → ω ∈ WOmni)
nninfdcinf.n (𝜑𝑁 ∈ ℕ)
Assertion
Ref Expression
nninfdcinf (𝜑DECID 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.
StepHypRef Expression
1 fveq1 5509 . . . . . 6 (𝑓 = 𝑁 → (𝑓𝑥) = (𝑁𝑥))
21eqeq1d 2186 . . . . 5 (𝑓 = 𝑁 → ((𝑓𝑥) = 1o ↔ (𝑁𝑥) = 1o))
32ralbidv 2477 . . . 4 (𝑓 = 𝑁 → (∀𝑥 ∈ ω (𝑓𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝑁𝑥) = 1o))
43dcbid 838 . . 3 (𝑓 = 𝑁 → (DECID𝑥 ∈ ω (𝑓𝑥) = 1oDECID𝑥 ∈ ω (𝑁𝑥) = 1o))
5 nninfdcinf.w . . . 4 (𝜑 → ω ∈ WOmni)
65elexd 2750 . . . . 5 (𝜑 → ω ∈ V)
7 iswomnimap 7157 . . . . 5 (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o𝑚 ω)DECID𝑥 ∈ ω (𝑓𝑥) = 1o))
86, 7syl 14 . . . 4 (𝜑 → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o𝑚 ω)DECID𝑥 ∈ ω (𝑓𝑥) = 1o))
95, 8mpbid 147 . . 3 (𝜑 → ∀𝑓 ∈ (2o𝑚 ω)DECID𝑥 ∈ ω (𝑓𝑥) = 1o)
10 nninfdcinf.n . . . . 5 (𝜑𝑁 ∈ ℕ)
11 nninff 7114 . . . . 5 (𝑁 ∈ ℕ𝑁:ω⟶2o)
1210, 11syl 14 . . . 4 (𝜑𝑁:ω⟶2o)
13 2onn 6515 . . . . . 6 2o ∈ ω
1413elexi 2749 . . . . 5 2o ∈ V
15 omex 4588 . . . . 5 ω ∈ V
1614, 15elmap 6670 . . . 4 (𝑁 ∈ (2o𝑚 ω) ↔ 𝑁:ω⟶2o)
1712, 16sylibr 134 . . 3 (𝜑𝑁 ∈ (2o𝑚 ω))
184, 9, 17rspcdva 2846 . 2 (𝜑DECID𝑥 ∈ ω (𝑁𝑥) = 1o)
1912ffnd 5361 . . . 4 (𝜑𝑁 Fn ω)
20 eqidd 2178 . . . 4 (𝑥 = 𝑖 → 1o = 1o)
21 1onn 6514 . . . . 5 1o ∈ ω
2221a1i 9 . . . 4 ((𝜑𝑥 ∈ ω) → 1o ∈ ω)
2321a1i 9 . . . 4 ((𝜑𝑖 ∈ ω) → 1o ∈ ω)
2419, 20, 22, 23fnmptfvd 5615 . . 3 (𝜑 → (𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝑁𝑥) = 1o))
2524dcbid 838 . 2 (𝜑 → (DECID 𝑁 = (𝑖 ∈ ω ↦ 1o) ↔ DECID𝑥 ∈ ω (𝑁𝑥) = 1o))
2618, 25mpbird 167 1 (𝜑DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737  cmpt 4061  ωcom 4585  wf 5207  cfv 5211  (class class class)co 5868  1oc1o 6403  2oc2o 6404  𝑚 cmap 6641  xnninf 7111  WOmnicwomni 7154
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1o 6410  df-2o 6411  df-map 6643  df-nninf 7112  df-womni 7155
This theorem is referenced by: (None)
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