Theorem nninfall 16774

Description: Given a decidable predicate on ℕ_∞, 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 1_o (which can be thought of as false and true) to every element of ℕ_∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)

Hypotheses

Ref	Expression
nninfall.q	⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
nninfall.inf	⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
nninfall.n	⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)

Assertion

Ref	Expression
nninfall	⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 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.

Step	Hyp	Ref	Expression
1		1n0 6664	. . . . 5 ⊢ 1o ≠ ∅
2	1	nesymi 2458	. . . 4 ⊢ ¬ ∅ = 1o
3		simplr 529	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → 𝑝 ∈ ℕ∞)
4		nninff 7412	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℕ∞ → 𝑝:ω⟶2o)
5	4	ffnd 5508	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℕ∞ → 𝑝 Fn ω)
6	3, 5	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → 𝑝 Fn ω)
7		nninfall.q	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
8	7	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
9		nninfall.inf	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
10	9	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)
11		nninfall.n	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
12	11	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
13		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → (𝑄‘𝑝) = ∅)
14	8, 10, 12, 3, 13	nninfalllem1 16773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → ∀𝑛 ∈ ω (𝑝‘𝑛) = 1o)
15		eqeq1 2239	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑝‘𝑛) → (𝑎 = 1o ↔ (𝑝‘𝑛) = 1o))
16	15	ralrn 5814	. . . . . . . . . . . . . 14 ⊢ (𝑝 Fn ω → (∀𝑎 ∈ ran 𝑝 𝑎 = 1o ↔ ∀𝑛 ∈ ω (𝑝‘𝑛) = 1o))
17	3, 5, 16	3syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → (∀𝑎 ∈ ran 𝑝 𝑎 = 1o ↔ ∀𝑛 ∈ ω (𝑝‘𝑛) = 1o))
18	14, 17	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → ∀𝑎 ∈ ran 𝑝 𝑎 = 1o)
19		peano1 4715	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ ω
20		elex2 2829	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ ω → ∃𝑏 𝑏 ∈ ω)
21	19, 20	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ∃𝑏 𝑏 ∈ ω
22		fdm 5513	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝:ω⟶2o → dom 𝑝 = ω)
23	22	eleq2d 2302	. . . . . . . . . . . . . . . 16 ⊢ (𝑝:ω⟶2o → (𝑏 ∈ dom 𝑝 ↔ 𝑏 ∈ ω))
24	23	exbidv 1874	. . . . . . . . . . . . . . 15 ⊢ (𝑝:ω⟶2o → (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑏 𝑏 ∈ ω))
25	21, 24	mpbiri 168	. . . . . . . . . . . . . 14 ⊢ (𝑝:ω⟶2o → ∃𝑏 𝑏 ∈ dom 𝑝)
26		dmmrnm 4975	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏 𝑏 ∈ dom 𝑝 ↔ ∃𝑎 𝑎 ∈ ran 𝑝)
27		eqsnm 3858	. . . . . . . . . . . . . . 15 ⊢ (∃𝑎 𝑎 ∈ ran 𝑝 → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
28	26, 27	sylbi 121	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 𝑏 ∈ dom 𝑝 → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
29	25, 28	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑝:ω⟶2o → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
30	3, 4, 29	3syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → (ran 𝑝 = {1o} ↔ ∀𝑎 ∈ ran 𝑝 𝑎 = 1o))
31	18, 30	mpbird 167	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → ran 𝑝 = {1o})
32		eqimss 3291	. . . . . . . . . . 11 ⊢ (ran 𝑝 = {1o} → ran 𝑝 ⊆ {1o})
33	31, 32	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → ran 𝑝 ⊆ {1o})
34		df-f 5355	. . . . . . . . . 10 ⊢ (𝑝:ω⟶{1o} ↔ (𝑝 Fn ω ∧ ran 𝑝 ⊆ {1o}))
35	6, 33, 34	sylanbrc 417	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → 𝑝:ω⟶{1o})
36		1onn 6752	. . . . . . . . . 10 ⊢ 1o ∈ ω
37		fconst2g 5898	. . . . . . . . . 10 ⊢ (1o ∈ ω → (𝑝:ω⟶{1o} ↔ 𝑝 = (ω × {1o})))
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ (𝑝:ω⟶{1o} ↔ 𝑝 = (ω × {1o}))
39	35, 38	sylib 122	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → 𝑝 = (ω × {1o}))
40		fconstmpt 4796	. . . . . . . 8 ⊢ (ω × {1o}) = (𝑥 ∈ ω ↦ 1o)
41	39, 40	eqtrdi 2281	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → 𝑝 = (𝑥 ∈ ω ↦ 1o))
42	41	fveq2d 5673	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → (𝑄‘𝑝) = (𝑄‘(𝑥 ∈ ω ↦ 1o)))
43	42, 13, 10	3eqtr3d 2273	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ ℕ∞) ∧ (𝑄‘𝑝) = ∅) → ∅ = 1o)
44	43	ex 115	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ℕ∞) → ((𝑄‘𝑝) = ∅ → ∅ = 1o))
45	2, 44	mtoi 670	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ ℕ∞) → ¬ (𝑄‘𝑝) = ∅)
46		elmapi 6903	. . . . . . 7 ⊢ (𝑄 ∈ (2o ↑𝑚 ℕ∞) → 𝑄:ℕ∞⟶2o)
47	7, 46	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑄:ℕ∞⟶2o)
48	47	ffvelcdmda 5811	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ℕ∞) → (𝑄‘𝑝) ∈ 2o)
49		elpri 3711	. . . . . 6 ⊢ ((𝑄‘𝑝) ∈ {∅, 1o} → ((𝑄‘𝑝) = ∅ ∨ (𝑄‘𝑝) = 1o))
50		df2o3 6661	. . . . . 6 ⊢ 2o = {∅, 1o}
51	49, 50	eleq2s 2327	. . . . 5 ⊢ ((𝑄‘𝑝) ∈ 2o → ((𝑄‘𝑝) = ∅ ∨ (𝑄‘𝑝) = 1o))
52	48, 51	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ℕ∞) → ((𝑄‘𝑝) = ∅ ∨ (𝑄‘𝑝) = 1o))
53	52	orcomd 737	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ ℕ∞) → ((𝑄‘𝑝) = 1o ∨ (𝑄‘𝑝) = ∅))
54	45, 53	ecased 1386	. 2 ⊢ ((𝜑 ∧ 𝑝 ∈ ℕ∞) → (𝑄‘𝑝) = 1o)
55	54	ralrimiva 2615	1 ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520   ⊆ wss 3210  ∅c0 3507  ifcif 3619  {csn 3688  {cpr 3689   ↦ cmpt 4170  ωcom 4711   × cxp 4746  dom cdm 4748  ran crn 4749   Fn wfn 5346  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  1_oc1o 6639  2_oc2o 6640   ↑_𝑚 cmap 6881  ℕ_∞xnninf 7409

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1o 6646  df-2o 6647  df-map 6883  df-nninf 7410

This theorem is referenced by:  nninfsel  16782  nninffeq  16785