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Theorem nninfall 15156
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 6451 . . . . 5 1o ≠ ∅
21nesymi 2406 . . . 4 ¬ ∅ = 1o
3 simplr 528 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 ∈ ℕ)
4 nninff 7139 . . . . . . . . . . . 12 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54ffnd 5381 . . . . . . . . . . 11 (𝑝 ∈ ℕ𝑝 Fn ω)
63, 5syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 Fn ω)
7 nninfall.q . . . . . . . . . . . . . . 15 (𝜑𝑄 ∈ (2o𝑚))
87ad2antrr 488 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑄 ∈ (2o𝑚))
9 nninfall.inf . . . . . . . . . . . . . . 15 (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
109ad2antrr 488 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
11 nninfall.n . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
1211ad2antrr 488 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
13 simpr 110 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = ∅)
148, 10, 12, 3, 13nninfalllem1 15155 . . . . . . . . . . . . 13 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑝𝑛) = 1o)
15 eqeq1 2196 . . . . . . . . . . . . . . 15 (𝑎 = (𝑝𝑛) → (𝑎 = 1o ↔ (𝑝𝑛) = 1o))
1615ralrn 5670 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . 12 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑎 ∈ ran 𝑝 𝑎 = 1o)
19 peano1 4608 . . . . . . . . . . . . . . . 16 ∅ ∈ ω
20 elex2 2768 . . . . . . . . . . . . . . . 16 (∅ ∈ ω → ∃𝑏 𝑏 ∈ ω)
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15 𝑏 𝑏 ∈ ω
22 fdm 5386 . . . . . . . . . . . . . . . . 17 (𝑝:ω⟶2o → dom 𝑝 = ω)
2322eleq2d 2259 . . . . . . . . . . . . . . . 16 (𝑝:ω⟶2o → (𝑏 ∈ dom 𝑝𝑏 ∈ ω))
2423exbidv 1836 . . . . . . . . . . . . . . 15 (𝑝:ω⟶2o → (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑏 𝑏 ∈ ω))
2521, 24mpbiri 168 . . . . . . . . . . . . . 14 (𝑝:ω⟶2o → ∃𝑏 𝑏 ∈ dom 𝑝)
26 dmmrnm 4861 . . . . . . . . . . . . . . 15 (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑎 𝑎 ∈ ran 𝑝)
27 eqsnm 3770 . . . . . . . . . . . . . . 15 (∃𝑎 𝑎 ∈ ran 𝑝 → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
2826, 27sylbi 121 . . . . . . . . . . . . . 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 167 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 = {1o})
32 eqimss 3224 . . . . . . . . . . 11 (ran 𝑝 = {1o} → ran 𝑝 ⊆ {1o})
3331, 32syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 ⊆ {1o})
34 df-f 5235 . . . . . . . . . 10 (𝑝:ω⟶{1o} ↔ (𝑝 Fn ω ∧ ran 𝑝 ⊆ {1o}))
356, 33, 34sylanbrc 417 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝:ω⟶{1o})
36 1onn 6539 . . . . . . . . . 10 1o ∈ ω
37 fconst2g 5747 . . . . . . . . . 10 (1o ∈ ω → (𝑝:ω⟶{1o} ↔ 𝑝 = (ω × {1o})))
3836, 37ax-mp 5 . . . . . . . . 9 (𝑝:ω⟶{1o} ↔ 𝑝 = (ω × {1o}))
3935, 38sylib 122 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (ω × {1o}))
40 fconstmpt 4688 . . . . . . . 8 (ω × {1o}) = (𝑥 ∈ ω ↦ 1o)
4139, 40eqtrdi 2238 . . . . . . 7 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (𝑥 ∈ ω ↦ 1o))
4241fveq2d 5534 . . . . . 6 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = (𝑄‘(𝑥 ∈ ω ↦ 1o)))
4342, 13, 103eqtr3d 2230 . . . . 5 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∅ = 1o)
4443ex 115 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ → ∅ = 1o))
452, 44mtoi 665 . . 3 ((𝜑𝑝 ∈ ℕ) → ¬ (𝑄𝑝) = ∅)
46 elmapi 6688 . . . . . . 7 (𝑄 ∈ (2o𝑚) → 𝑄:ℕ⟶2o)
477, 46syl 14 . . . . . 6 (𝜑𝑄:ℕ⟶2o)
4847ffvelcdmda 5667 . . . . 5 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) ∈ 2o)
49 elpri 3630 . . . . . 6 ((𝑄𝑝) ∈ {∅, 1o} → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
50 df2o3 6449 . . . . . 6 2o = {∅, 1o}
5149, 50eleq2s 2284 . . . . 5 ((𝑄𝑝) ∈ 2o → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
5248, 51syl 14 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
5352orcomd 730 . . 3 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = 1o ∨ (𝑄𝑝) = ∅))
5445, 53ecased 1360 . 2 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) = 1o)
5554ralrimiva 2563 1 (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wex 1503  wcel 2160  wral 2468  wss 3144  c0 3437  ifcif 3549  {csn 3607  {cpr 3608  cmpt 4079  ωcom 4604   × cxp 4639  dom cdm 4641  ran crn 4642   Fn wfn 5226  wf 5227  cfv 5231  (class class class)co 5891  1oc1o 6428  2oc2o 6429  𝑚 cmap 6666  xnninf 7136
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1o 6435  df-2o 6436  df-map 6668  df-nninf 7137
This theorem is referenced by:  nninfsel  15164  nninffeq  15167
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