Theorem nnnninf 7023

Description: Elements of ℕ_∞ corresponding to natural numbers. The natural number 𝑁 corresponds to a sequence of 𝑁 ones followed by zeroes. Contrast to a sequence which is all ones as seen at infnninf 7022. Remark/TODO: the theorem still holds if 𝑁 = ω, that is, the antecedent could be weakened to 𝑁 ∈ suc ω. (Contributed by Jim Kingdon, 14-Jul-2022.)

Assertion

Ref	Expression
nnnninf	⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)

Distinct variable group:   𝑖,𝑁

Proof of Theorem nnnninf

Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 6321	. . . . . . . 8 ⊢ 1o ∈ V
2	1	sucid 4339	. . . . . . 7 ⊢ 1o ∈ suc 1o
3		df-2o 6314	. . . . . . 7 ⊢ 2o = suc 1o
4	2, 3	eleqtrri 2215	. . . . . 6 ⊢ 1o ∈ 2o
5	4	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
6		2on0 6323	. . . . . . 7 ⊢ 2o ≠ ∅
7		2onn 6417	. . . . . . . 8 ⊢ 2o ∈ ω
8		nn0eln0 4533	. . . . . . . 8 ⊢ (2o ∈ ω → (∅ ∈ 2o ↔ 2o ≠ ∅))
9	7, 8	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ 2o ↔ 2o ≠ ∅)
10	6, 9	mpbir 145	. . . . . 6 ⊢ ∅ ∈ 2o
11	10	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
12		nndcel 6396	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑖 ∈ 𝑁)
13	12	ancoms 266	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → DECID 𝑖 ∈ 𝑁)
14	5, 11, 13	ifcldcd 3507	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → if(𝑖 ∈ 𝑁, 1o, ∅) ∈ 2o)
15		eqid 2139	. . . 4 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
16	14, 15	fmptd 5574	. . 3 ⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)):ω⟶2o)
17	7	elexi 2698	. . . 4 ⊢ 2o ∈ V
18		omex 4507	. . . 4 ⊢ ω ∈ V
19	17, 18	elmap 6571	. . 3 ⊢ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)):ω⟶2o)
20	16, 19	sylibr 133	. 2 ⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ (2o ↑𝑚 ω))
21		ssid 3117	. . . . . . . . 9 ⊢ 1o ⊆ 1o
22		iftrue 3479	. . . . . . . . . . 11 ⊢ (suc 𝑗 ∈ 𝑁 → if(suc 𝑗 ∈ 𝑁, 1o, ∅) = 1o)
23	22	sseq1d 3126	. . . . . . . . . 10 ⊢ (suc 𝑗 ∈ 𝑁 → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ 1o ⊆ 1o))
24	23	adantl 275	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ suc 𝑗 ∈ 𝑁) → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ 1o ⊆ 1o))
25	21, 24	mpbiri 167	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ suc 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
26		0ss 3401	. . . . . . . . 9 ⊢ ∅ ⊆ 1o
27		iffalse 3482	. . . . . . . . . . 11 ⊢ (¬ suc 𝑗 ∈ 𝑁 → if(suc 𝑗 ∈ 𝑁, 1o, ∅) = ∅)
28	27	sseq1d 3126	. . . . . . . . . 10 ⊢ (¬ suc 𝑗 ∈ 𝑁 → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ ∅ ⊆ 1o))
29	28	adantl 275	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ suc 𝑗 ∈ 𝑁) → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ ∅ ⊆ 1o))
30	26, 29	mpbiri 167	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ suc 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
31		peano2 4509	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
32	31	adantl 275	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → suc 𝑗 ∈ ω)
33		simpl 108	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → 𝑁 ∈ ω)
34		nndcel 6396	. . . . . . . . . 10 ⊢ ((suc 𝑗 ∈ ω ∧ 𝑁 ∈ ω) → DECID suc 𝑗 ∈ 𝑁)
35	32, 33, 34	syl2anc 408	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → DECID suc 𝑗 ∈ 𝑁)
36		exmiddc 821	. . . . . . . . 9 ⊢ (DECID suc 𝑗 ∈ 𝑁 → (suc 𝑗 ∈ 𝑁 ∨ ¬ suc 𝑗 ∈ 𝑁))
37	35, 36	syl 14	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (suc 𝑗 ∈ 𝑁 ∨ ¬ suc 𝑗 ∈ 𝑁))
38	25, 30, 37	mpjaodan 787	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
39	38	adantr 274	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
40		iftrue 3479	. . . . . . 7 ⊢ (𝑗 ∈ 𝑁 → if(𝑗 ∈ 𝑁, 1o, ∅) = 1o)
41	40	adantl 275	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑗 ∈ 𝑁) → if(𝑗 ∈ 𝑁, 1o, ∅) = 1o)
42	39, 41	sseqtrrd 3136	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ if(𝑗 ∈ 𝑁, 1o, ∅))
43		ssid 3117	. . . . . . 7 ⊢ ∅ ⊆ ∅
44	43	a1i 9	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → ∅ ⊆ ∅)
45		nnord 4525	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
46		ordtr 4300	. . . . . . . . . . . 12 ⊢ (Ord 𝑁 → Tr 𝑁)
47	45, 46	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → Tr 𝑁)
48		trsuc 4344	. . . . . . . . . . 11 ⊢ ((Tr 𝑁 ∧ suc 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
49	47, 48	sylan 281	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ suc 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
50	49	ex 114	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (suc 𝑗 ∈ 𝑁 → 𝑗 ∈ 𝑁))
51	50	adantr 274	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (suc 𝑗 ∈ 𝑁 → 𝑗 ∈ 𝑁))
52	51	con3dimp 624	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → ¬ suc 𝑗 ∈ 𝑁)
53	52, 27	syl 14	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) = ∅)
54		iffalse 3482	. . . . . . 7 ⊢ (¬ 𝑗 ∈ 𝑁 → if(𝑗 ∈ 𝑁, 1o, ∅) = ∅)
55	54	adantl 275	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → if(𝑗 ∈ 𝑁, 1o, ∅) = ∅)
56	44, 53, 55	3sstr4d 3142	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ if(𝑗 ∈ 𝑁, 1o, ∅))
57		nndcel 6396	. . . . . . 7 ⊢ ((𝑗 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑗 ∈ 𝑁)
58	57	ancoms 266	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → DECID 𝑗 ∈ 𝑁)
59		exmiddc 821	. . . . . 6 ⊢ (DECID 𝑗 ∈ 𝑁 → (𝑗 ∈ 𝑁 ∨ ¬ 𝑗 ∈ 𝑁))
60	58, 59	syl 14	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (𝑗 ∈ 𝑁 ∨ ¬ 𝑗 ∈ 𝑁))
61	42, 56, 60	mpjaodan 787	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ if(𝑗 ∈ 𝑁, 1o, ∅))
62	4	a1i 9	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → 1o ∈ 2o)
63	10	a1i 9	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ 2o)
64	62, 63, 35	ifcldcd 3507	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o)
65		eleq1 2202	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑁 ↔ suc 𝑗 ∈ 𝑁))
66	65	ifbid 3493	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(suc 𝑗 ∈ 𝑁, 1o, ∅))
67	66, 15	fvmptg 5497	. . . . 5 ⊢ ((suc 𝑗 ∈ ω ∧ if(suc 𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) = if(suc 𝑗 ∈ 𝑁, 1o, ∅))
68	32, 64, 67	syl2anc 408	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) = if(suc 𝑗 ∈ 𝑁, 1o, ∅))
69		simpr 109	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
70	62, 63, 58	ifcldcd 3507	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o)
71		eleq1 2202	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝑁 ↔ 𝑗 ∈ 𝑁))
72	71	ifbid 3493	. . . . . 6 ⊢ (𝑖 = 𝑗 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(𝑗 ∈ 𝑁, 1o, ∅))
73	72, 15	fvmptg 5497	. . . . 5 ⊢ ((𝑗 ∈ ω ∧ if(𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗) = if(𝑗 ∈ 𝑁, 1o, ∅))
74	69, 70, 73	syl2anc 408	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗) = if(𝑗 ∈ 𝑁, 1o, ∅))
75	61, 68, 74	3sstr4d 3142	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗))
76	75	ralrimiva 2505	. 2 ⊢ (𝑁 ∈ ω → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗))
77		fveq1 5420	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗))
78		fveq1 5420	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗))
79	77, 78	sseq12d 3128	. . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗)))
80	79	ralbidv 2437	. . 3 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗)))
81		df-nninf 7007	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
82	80, 81	elrab2 2843	. 2 ⊢ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞ ↔ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗)))
83	20, 76, 82	sylanbrc 413	1 ⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416   ⊆ wss 3071  ∅c0 3363  ifcif 3474   ↦ cmpt 3989  Tr wtr 4026  Ord word 4284  suc csuc 4287  ωcom 4504  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  2_oc2o 6307   ↑_𝑚 cmap 6542  ℕ_∞xnninf 7005

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1o 6313  df-2o 6314  df-map 6544  df-nninf 7007

This theorem is referenced by:  fnn0nninf  10210  nninfsellemdc  13206  nninfsellemqall  13211  nninffeq  13216