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Theorem nninfall 11557
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 6179 . . . . 5 1𝑜 ≠ ∅
21nesymi 2301 . . . 4 ¬ ∅ = 1𝑜
3 simplr 497 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 ∈ ℕ)
4 nninff 11551 . . . . . . . . . . . 12 (𝑝 ∈ ℕ𝑝:ω⟶2𝑜)
54ffnd 5148 . . . . . . . . . . 11 (𝑝 ∈ ℕ𝑝 Fn ω)
63, 5syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 Fn ω)
7 nninfall.q . . . . . . . . . . . . . . 15 (𝜑𝑄 ∈ (2𝑜𝑚))
87ad2antrr 472 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑄 ∈ (2𝑜𝑚))
9 nninfall.inf . . . . . . . . . . . . . . 15 (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1𝑜)) = 1𝑜)
109ad2antrr 472 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄‘(𝑥 ∈ ω ↦ 1𝑜)) = 1𝑜)
11 nninfall.n . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1𝑜, ∅))) = 1𝑜)
1211ad2antrr 472 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1𝑜, ∅))) = 1𝑜)
13 simpr 108 . . . . . . . . . . . . . 14 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = ∅)
148, 10, 12, 3, 13nninfalllem1 11556 . . . . . . . . . . . . 13 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑛 ∈ ω (𝑝𝑛) = 1𝑜)
15 eqeq1 2094 . . . . . . . . . . . . . . 15 (𝑎 = (𝑝𝑛) → (𝑎 = 1𝑜 ↔ (𝑝𝑛) = 1𝑜))
1615ralrn 5421 . . . . . . . . . . . . . 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 165 . . . . . . . . . . . 12 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∀𝑎 ∈ ran 𝑝 𝑎 = 1𝑜)
19 peano1 4399 . . . . . . . . . . . . . . . 16 ∅ ∈ ω
20 elex2 2635 . . . . . . . . . . . . . . . 16 (∅ ∈ ω → ∃𝑏 𝑏 ∈ ω)
2119, 20ax-mp 7 . . . . . . . . . . . . . . 15 𝑏 𝑏 ∈ ω
22 fdm 5152 . . . . . . . . . . . . . . . . 17 (𝑝:ω⟶2𝑜 → dom 𝑝 = ω)
2322eleq2d 2157 . . . . . . . . . . . . . . . 16 (𝑝:ω⟶2𝑜 → (𝑏 ∈ dom 𝑝𝑏 ∈ ω))
2423exbidv 1753 . . . . . . . . . . . . . . 15 (𝑝:ω⟶2𝑜 → (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑏 𝑏 ∈ ω))
2521, 24mpbiri 166 . . . . . . . . . . . . . 14 (𝑝:ω⟶2𝑜 → ∃𝑏 𝑏 ∈ dom 𝑝)
26 dmmrnm 4643 . . . . . . . . . . . . . . 15 (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑎 𝑎 ∈ ran 𝑝)
27 eqsnm 3594 . . . . . . . . . . . . . . 15 (∃𝑎 𝑎 ∈ ran 𝑝 → (ran 𝑝 = {1𝑜} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1𝑜))
2826, 27sylbi 119 . . . . . . . . . . . . . 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 165 . . . . . . . . . . 11 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 = {1𝑜})
32 eqimss 3076 . . . . . . . . . . 11 (ran 𝑝 = {1𝑜} → ran 𝑝 ⊆ {1𝑜})
3331, 32syl 14 . . . . . . . . . 10 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ran 𝑝 ⊆ {1𝑜})
34 df-f 5006 . . . . . . . . . 10 (𝑝:ω⟶{1𝑜} ↔ (𝑝 Fn ω ∧ ran 𝑝 ⊆ {1𝑜}))
356, 33, 34sylanbrc 408 . . . . . . . . 9 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝:ω⟶{1𝑜})
36 1onn 6259 . . . . . . . . . 10 1𝑜 ∈ ω
37 fconst2g 5494 . . . . . . . . . 10 (1𝑜 ∈ ω → (𝑝:ω⟶{1𝑜} ↔ 𝑝 = (ω × {1𝑜})))
3836, 37ax-mp 7 . . . . . . . . 9 (𝑝:ω⟶{1𝑜} ↔ 𝑝 = (ω × {1𝑜}))
3935, 38sylib 120 . . . . . . . 8 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (ω × {1𝑜}))
40 fconstmpt 4473 . . . . . . . 8 (ω × {1𝑜}) = (𝑥 ∈ ω ↦ 1𝑜)
4139, 40syl6eq 2136 . . . . . . 7 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → 𝑝 = (𝑥 ∈ ω ↦ 1𝑜))
4241fveq2d 5293 . . . . . 6 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → (𝑄𝑝) = (𝑄‘(𝑥 ∈ ω ↦ 1𝑜)))
4342, 13, 103eqtr3d 2128 . . . . 5 (((𝜑𝑝 ∈ ℕ) ∧ (𝑄𝑝) = ∅) → ∅ = 1𝑜)
4443ex 113 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ → ∅ = 1𝑜))
452, 44mtoi 625 . . 3 ((𝜑𝑝 ∈ ℕ) → ¬ (𝑄𝑝) = ∅)
46 elmapi 6407 . . . . . . 7 (𝑄 ∈ (2𝑜𝑚) → 𝑄:ℕ⟶2𝑜)
477, 46syl 14 . . . . . 6 (𝜑𝑄:ℕ⟶2𝑜)
4847ffvelrnda 5418 . . . . 5 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) ∈ 2𝑜)
49 elpri 3464 . . . . . 6 ((𝑄𝑝) ∈ {∅, 1𝑜} → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1𝑜))
50 df2o3 6177 . . . . . 6 2𝑜 = {∅, 1𝑜}
5149, 50eleq2s 2182 . . . . 5 ((𝑄𝑝) ∈ 2𝑜 → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1𝑜))
5248, 51syl 14 . . . 4 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = ∅ ∨ (𝑄𝑝) = 1𝑜))
5352orcomd 683 . . 3 ((𝜑𝑝 ∈ ℕ) → ((𝑄𝑝) = 1𝑜 ∨ (𝑄𝑝) = ∅))
5445, 53ecased 1285 . 2 ((𝜑𝑝 ∈ ℕ) → (𝑄𝑝) = 1𝑜)
5554ralrimiva 2446 1 (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 664   = wceq 1289  wex 1426  wcel 1438  wral 2359  wss 2997  c0 3284  ifcif 3389  {csn 3441  {cpr 3442  cmpt 3891  ωcom 4395   × cxp 4426  dom cdm 4428  ran crn 4429   Fn wfn 4997  wf 4998  cfv 5002  (class class class)co 5634  1𝑜c1o 6156  2𝑜c2o 6157  𝑚 cmap 6385  xnninf 6768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770
This theorem is referenced by:  nninfsel  11566
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