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Theorem nninfall 13131
Description: Given a decidable predicate on , showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which 𝑄 is a decidable predicate is that it assigns a value of either or 1o (which can be thought of as false and true) to every element of . Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
Hypotheses
Ref Expression
nninfall.q (𝜑𝑄 ∈ (2o𝑚))
nninfall.inf (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
nninfall.n (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
Assertion
Ref Expression
nninfall (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)
Distinct variable groups:   𝑄,𝑛,𝑖   𝑛,𝑝,𝑖,𝜑
Allowed substitution hints:   𝜑(𝑥)   𝑄(𝑥,𝑝)

Proof of Theorem nninfall
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 6297 . . . . 5 1o ≠ ∅
21nesymi 2331 . . . 4 ¬ ∅ = 1o
3 simplr 504 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 ∈ ℕ)
4 nninff 13125 . . . . . . . . . . . 12 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54ffnd 5243 . . . . . . . . . . 11 (𝑝 ∈ ℕ𝑝 Fn ω)
63, 5syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 Fn ω)
7 nninfall.q . . . . . . . . . . . . . . 15 (𝜑𝑄 ∈ (2o𝑚))
87ad2antrr 479 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑄 ∈ (2o𝑚))
9 nninfall.inf . . . . . . . . . . . . . . 15 (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
109ad2antrr 479 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
11 nninfall.n . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
1211ad2antrr 479 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
13 simpr 109 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = ∅)
148, 10, 12, 3, 13nninfalllem1 13130 . . . . . . . . . . . . 13 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑝𝑛) = 1o)
15 eqeq1 2124 . . . . . . . . . . . . . . 15 (𝑎 = (𝑝𝑛) → (𝑎 = 1o ↔ (𝑝𝑛) = 1o))
1615ralrn 5526 . . . . . . . . . . . . . 14 (𝑝 Fn ω → (∀𝑎 ∈ ran 𝑝 𝑎 = 1o ↔ ∀𝑛 ∈ ω (𝑝𝑛) = 1o))
173, 5, 163syl 17 . . . . . . . . . . . . 13 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (∀𝑎 ∈ ran 𝑝 𝑎 = 1o ↔ ∀𝑛 ∈ ω (𝑝𝑛) = 1o))
1814, 17mpbird 166 . . . . . . . . . . . 12 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑎 ∈ ran 𝑝 𝑎 = 1o)
19 peano1 4478 . . . . . . . . . . . . . . . 16 ∅ ∈ ω
20 elex2 2676 . . . . . . . . . . . . . . . 16 (∅ ∈ ω → ∃𝑏 𝑏 ∈ ω)
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15 𝑏 𝑏 ∈ ω
22 fdm 5248 . . . . . . . . . . . . . . . . 17 (𝑝:ω⟶2o → dom 𝑝 = ω)
2322eleq2d 2187 . . . . . . . . . . . . . . . 16 (𝑝:ω⟶2o → (𝑏 ∈ dom 𝑝𝑏 ∈ ω))
2423exbidv 1781 . . . . . . . . . . . . . . 15 (𝑝:ω⟶2o → (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑏 𝑏 ∈ ω))
2521, 24mpbiri 167 . . . . . . . . . . . . . 14 (𝑝:ω⟶2o → ∃𝑏 𝑏 ∈ dom 𝑝)
26 dmmrnm 4728 . . . . . . . . . . . . . . 15 (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑎 𝑎 ∈ ran 𝑝)
27 eqsnm 3652 . . . . . . . . . . . . . . 15 (∃𝑎 𝑎 ∈ ran 𝑝 → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
2826, 27sylbi 120 . . . . . . . . . . . . . 14 (∃𝑏 𝑏 ∈ dom 𝑝 → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
2925, 28syl 14 . . . . . . . . . . . . 13 (𝑝:ω⟶2o → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
303, 4, 293syl 17 . . . . . . . . . . . 12 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
3118, 30mpbird 166 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 = {1o})
32 eqimss 3121 . . . . . . . . . . 11 (ran 𝑝 = {1o} → ran 𝑝 ⊆ {1o})
3331, 32syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 ⊆ {1o})
34 df-f 5097 . . . . . . . . . 10 (𝑝:ω⟶{1o} ↔ (𝑝 Fn ω ∧ ran 𝑝 ⊆ {1o}))
356, 33, 34sylanbrc 413 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝:ω⟶{1o})
36 1onn 6384 . . . . . . . . . 10 1o ∈ ω
37 fconst2g 5603 . . . . . . . . . 10 (1o ∈ ω → (𝑝:ω⟶{1o} ↔ 𝑝 = (ω × {1o})))
3836, 37ax-mp 5 . . . . . . . . 9 (𝑝:ω⟶{1o} ↔ 𝑝 = (ω × {1o}))
3935, 38sylib 121 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (ω × {1o}))
40 fconstmpt 4556 . . . . . . . 8 (ω × {1o}) = (𝑥 ∈ ω ↦ 1o)
4139, 40syl6eq 2166 . . . . . . 7 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (𝑥 ∈ ω ↦ 1o))
4241fveq2d 5393 . . . . . 6 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = (𝑄‘(𝑥 ∈ ω ↦ 1o)))
4342, 13, 103eqtr3d 2158 . . . . 5 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∅ = 1o)
4443ex 114 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ → ∅ = 1o))
452, 44mtoi 638 . . 3 ((𝜑𝑝 ∈ ℕ) → ¬ (𝑄𝑝) = ∅)
46 elmapi 6532 . . . . . . 7 (𝑄 ∈ (2o𝑚) → 𝑄:ℕ⟶2o)
477, 46syl 14 . . . . . 6 (𝜑𝑄:ℕ⟶2o)
4847ffvelrnda 5523 . . . . 5 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) ∈ 2o)
49 elpri 3520 . . . . . 6 ((𝑄𝑝) ∈ {∅, 1o} → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
50 df2o3 6295 . . . . . 6 2o = {∅, 1o}
5149, 50eleq2s 2212 . . . . 5 ((𝑄𝑝) ∈ 2o → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
5248, 51syl 14 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
5352orcomd 703 . . 3 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = 1o ∨ (𝑄𝑝) = ∅))
5445, 53ecased 1312 . 2 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) = 1o)
5554ralrimiva 2482 1 (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 682   = wceq 1316  wex 1453  wcel 1465  wral 2393  wss 3041  c0 3333  ifcif 3444  {csn 3497  {cpr 3498  cmpt 3959  ωcom 4474   × cxp 4507  dom cdm 4509  ran crn 4510   Fn wfn 5088  wf 5089  cfv 5093  (class class class)co 5742  1oc1o 6274  2oc2o 6275  𝑚 cmap 6510  xnninf 6973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1o 6281  df-2o 6282  df-map 6512  df-nninf 6975
This theorem is referenced by:  nninfsel  13140  nninffeq  13143
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