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Theorem nninfall 14042
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 6411 . . . . 5 1o ≠ ∅
21nesymi 2386 . . . 4 ¬ ∅ = 1o
3 simplr 525 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 ∈ ℕ)
4 nninff 7099 . . . . . . . . . . . 12 (𝑝 ∈ ℕ𝑝:ω⟶2o)
54ffnd 5348 . . . . . . . . . . 11 (𝑝 ∈ ℕ𝑝 Fn ω)
63, 5syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 Fn ω)
7 nninfall.q . . . . . . . . . . . . . . 15 (𝜑𝑄 ∈ (2o𝑚))
87ad2antrr 485 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑄 ∈ (2o𝑚))
9 nninfall.inf . . . . . . . . . . . . . . 15 (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
109ad2antrr 485 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
11 nninfall.n . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)
1211ad2antrr 485 . . . . . . . . . . . . . 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 14041 . . . . . . . . . . . . 13 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑝𝑛) = 1o)
15 eqeq1 2177 . . . . . . . . . . . . . . 15 (𝑎 = (𝑝𝑛) → (𝑎 = 1o ↔ (𝑝𝑛) = 1o))
1615ralrn 5634 . . . . . . . . . . . . . 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 4578 . . . . . . . . . . . . . . . 16 ∅ ∈ ω
20 elex2 2746 . . . . . . . . . . . . . . . 16 (∅ ∈ ω → ∃𝑏 𝑏 ∈ ω)
2119, 20ax-mp 5 . . . . . . . . . . . . . . 15 𝑏 𝑏 ∈ ω
22 fdm 5353 . . . . . . . . . . . . . . . . 17 (𝑝:ω⟶2o → dom 𝑝 = ω)
2322eleq2d 2240 . . . . . . . . . . . . . . . 16 (𝑝:ω⟶2o → (𝑏 ∈ dom 𝑝𝑏 ∈ ω))
2423exbidv 1818 . . . . . . . . . . . . . . 15 (𝑝:ω⟶2o → (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑏 𝑏 ∈ ω))
2521, 24mpbiri 167 . . . . . . . . . . . . . 14 (𝑝:ω⟶2o → ∃𝑏 𝑏 ∈ dom 𝑝)
26 dmmrnm 4830 . . . . . . . . . . . . . . 15 (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑎 𝑎 ∈ ran 𝑝)
27 eqsnm 3742 . . . . . . . . . . . . . . 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 3201 . . . . . . . . . . 11 (ran 𝑝 = {1o} → ran 𝑝 ⊆ {1o})
3331, 32syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 ⊆ {1o})
34 df-f 5202 . . . . . . . . . 10 (𝑝:ω⟶{1o} ↔ (𝑝 Fn ω ∧ ran 𝑝 ⊆ {1o}))
356, 33, 34sylanbrc 415 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝:ω⟶{1o})
36 1onn 6499 . . . . . . . . . 10 1o ∈ ω
37 fconst2g 5711 . . . . . . . . . 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 4658 . . . . . . . 8 (ω × {1o}) = (𝑥 ∈ ω ↦ 1o)
4139, 40eqtrdi 2219 . . . . . . 7 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (𝑥 ∈ ω ↦ 1o))
4241fveq2d 5500 . . . . . 6 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = (𝑄‘(𝑥 ∈ ω ↦ 1o)))
4342, 13, 103eqtr3d 2211 . . . . 5 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∅ = 1o)
4443ex 114 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ → ∅ = 1o))
452, 44mtoi 659 . . 3 ((𝜑𝑝 ∈ ℕ) → ¬ (𝑄𝑝) = ∅)
46 elmapi 6648 . . . . . . 7 (𝑄 ∈ (2o𝑚) → 𝑄:ℕ⟶2o)
477, 46syl 14 . . . . . 6 (𝜑𝑄:ℕ⟶2o)
4847ffvelrnda 5631 . . . . 5 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) ∈ 2o)
49 elpri 3606 . . . . . 6 ((𝑄𝑝) ∈ {∅, 1o} → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
50 df2o3 6409 . . . . . 6 2o = {∅, 1o}
5149, 50eleq2s 2265 . . . . 5 ((𝑄𝑝) ∈ 2o → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
5248, 51syl 14 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1o))
5352orcomd 724 . . 3 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = 1o ∨ (𝑄𝑝) = ∅))
5445, 53ecased 1344 . 2 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) = 1o)
5554ralrimiva 2543 1 (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 703   = wceq 1348  wex 1485  wcel 2141  wral 2448  wss 3121  c0 3414  ifcif 3526  {csn 3583  {cpr 3584  cmpt 4050  ωcom 4574   × cxp 4609  dom cdm 4611  ran crn 4612   Fn wfn 5193  wf 5194  cfv 5198  (class class class)co 5853  1oc1o 6388  2oc2o 6389  𝑚 cmap 6626  xnninf 7096
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628  df-nninf 7097
This theorem is referenced by:  nninfsel  14050  nninffeq  14053
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