Theorem nnnninf 7301

Description: Elements of ℕ_∞ corresponding to natural numbers. The natural number 𝑁 corresponds to a sequence of 𝑁 ones followed by zeroes. This can be strengthened to include infinity, see nnnninf2 7302. (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		1lt2o 6596	. . . . . 6 ⊢ 1o ∈ 2o
2	1	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
3		0lt2o 6595	. . . . . 6 ⊢ ∅ ∈ 2o
4	3	a1i 9	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
5		nndcel 6654	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑖 ∈ 𝑁)
6	5	ancoms 268	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → DECID 𝑖 ∈ 𝑁)
7	2, 4, 6	ifcldcd 3640	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑖 ∈ ω) → if(𝑖 ∈ 𝑁, 1o, ∅) ∈ 2o)
8	7	fmpttd 5792	. . 3 ⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)):ω⟶2o)
9		2onn 6675	. . . . 5 ⊢ 2o ∈ ω
10	9	elexi 2812	. . . 4 ⊢ 2o ∈ V
11		omex 4685	. . . 4 ⊢ ω ∈ V
12	10, 11	elmap 6832	. . 3 ⊢ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)):ω⟶2o)
13	8, 12	sylibr 134	. 2 ⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ (2o ↑𝑚 ω))
14		ssid 3244	. . . . . . . . 9 ⊢ 1o ⊆ 1o
15		iftrue 3607	. . . . . . . . . . 11 ⊢ (suc 𝑗 ∈ 𝑁 → if(suc 𝑗 ∈ 𝑁, 1o, ∅) = 1o)
16	15	sseq1d 3253	. . . . . . . . . 10 ⊢ (suc 𝑗 ∈ 𝑁 → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ 1o ⊆ 1o))
17	16	adantl 277	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ suc 𝑗 ∈ 𝑁) → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ 1o ⊆ 1o))
18	14, 17	mpbiri 168	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ suc 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
19		0ss 3530	. . . . . . . . 9 ⊢ ∅ ⊆ 1o
20		iffalse 3610	. . . . . . . . . . 11 ⊢ (¬ suc 𝑗 ∈ 𝑁 → if(suc 𝑗 ∈ 𝑁, 1o, ∅) = ∅)
21	20	sseq1d 3253	. . . . . . . . . 10 ⊢ (¬ suc 𝑗 ∈ 𝑁 → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ ∅ ⊆ 1o))
22	21	adantl 277	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ suc 𝑗 ∈ 𝑁) → (if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o ↔ ∅ ⊆ 1o))
23	19, 22	mpbiri 168	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ suc 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
24		peano2 4687	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
25	24	adantl 277	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → suc 𝑗 ∈ ω)
26		simpl 109	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → 𝑁 ∈ ω)
27		nndcel 6654	. . . . . . . . . 10 ⊢ ((suc 𝑗 ∈ ω ∧ 𝑁 ∈ ω) → DECID suc 𝑗 ∈ 𝑁)
28	25, 26, 27	syl2anc 411	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → DECID suc 𝑗 ∈ 𝑁)
29		exmiddc 841	. . . . . . . . 9 ⊢ (DECID suc 𝑗 ∈ 𝑁 → (suc 𝑗 ∈ 𝑁 ∨ ¬ suc 𝑗 ∈ 𝑁))
30	28, 29	syl 14	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (suc 𝑗 ∈ 𝑁 ∨ ¬ suc 𝑗 ∈ 𝑁))
31	18, 23, 30	mpjaodan 803	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
32	31	adantr 276	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ 1o)
33		iftrue 3607	. . . . . . 7 ⊢ (𝑗 ∈ 𝑁 → if(𝑗 ∈ 𝑁, 1o, ∅) = 1o)
34	33	adantl 277	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑗 ∈ 𝑁) → if(𝑗 ∈ 𝑁, 1o, ∅) = 1o)
35	32, 34	sseqtrrd 3263	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ if(𝑗 ∈ 𝑁, 1o, ∅))
36		ssid 3244	. . . . . . 7 ⊢ ∅ ⊆ ∅
37	36	a1i 9	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → ∅ ⊆ ∅)
38		nnord 4704	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
39		ordtr 4469	. . . . . . . . . . . 12 ⊢ (Ord 𝑁 → Tr 𝑁)
40	38, 39	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → Tr 𝑁)
41		trsuc 4513	. . . . . . . . . . 11 ⊢ ((Tr 𝑁 ∧ suc 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
42	40, 41	sylan 283	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ suc 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
43	42	ex 115	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (suc 𝑗 ∈ 𝑁 → 𝑗 ∈ 𝑁))
44	43	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (suc 𝑗 ∈ 𝑁 → 𝑗 ∈ 𝑁))
45	44	con3dimp 638	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → ¬ suc 𝑗 ∈ 𝑁)
46	45, 20	syl 14	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) = ∅)
47		iffalse 3610	. . . . . . 7 ⊢ (¬ 𝑗 ∈ 𝑁 → if(𝑗 ∈ 𝑁, 1o, ∅) = ∅)
48	47	adantl 277	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → if(𝑗 ∈ 𝑁, 1o, ∅) = ∅)
49	37, 46, 48	3sstr4d 3269	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝑗 ∈ ω) ∧ ¬ 𝑗 ∈ 𝑁) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ if(𝑗 ∈ 𝑁, 1o, ∅))
50		nndcel 6654	. . . . . . 7 ⊢ ((𝑗 ∈ ω ∧ 𝑁 ∈ ω) → DECID 𝑗 ∈ 𝑁)
51	50	ancoms 268	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → DECID 𝑗 ∈ 𝑁)
52		exmiddc 841	. . . . . 6 ⊢ (DECID 𝑗 ∈ 𝑁 → (𝑗 ∈ 𝑁 ∨ ¬ 𝑗 ∈ 𝑁))
53	51, 52	syl 14	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → (𝑗 ∈ 𝑁 ∨ ¬ 𝑗 ∈ 𝑁))
54	35, 49, 53	mpjaodan 803	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ⊆ if(𝑗 ∈ 𝑁, 1o, ∅))
55	1	a1i 9	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → 1o ∈ 2o)
56	3	a1i 9	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ∅ ∈ 2o)
57	55, 56, 28	ifcldcd 3640	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(suc 𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o)
58		eleq1 2292	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑁 ↔ suc 𝑗 ∈ 𝑁))
59	58	ifbid 3624	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(suc 𝑗 ∈ 𝑁, 1o, ∅))
60		eqid 2229	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
61	59, 60	fvmptg 5712	. . . . 5 ⊢ ((suc 𝑗 ∈ ω ∧ if(suc 𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) = if(suc 𝑗 ∈ 𝑁, 1o, ∅))
62	25, 57, 61	syl2anc 411	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) = if(suc 𝑗 ∈ 𝑁, 1o, ∅))
63		simpr 110	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
64	55, 56, 51	ifcldcd 3640	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → if(𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o)
65		eleq1 2292	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝑁 ↔ 𝑗 ∈ 𝑁))
66	65	ifbid 3624	. . . . . 6 ⊢ (𝑖 = 𝑗 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(𝑗 ∈ 𝑁, 1o, ∅))
67	66, 60	fvmptg 5712	. . . . 5 ⊢ ((𝑗 ∈ ω ∧ if(𝑗 ∈ 𝑁, 1o, ∅) ∈ 2o) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗) = if(𝑗 ∈ 𝑁, 1o, ∅))
68	63, 64, 67	syl2anc 411	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗) = if(𝑗 ∈ 𝑁, 1o, ∅))
69	54, 62, 68	3sstr4d 3269	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗))
70	69	ralrimiva 2603	. 2 ⊢ (𝑁 ∈ ω → ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗))
71		fveq1 5628	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘suc 𝑗))
72		fveq1 5628	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑗))
73	71, 72	sseq12d 3255	. . . 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)))
74	73	ralbidv 2530	. . 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)))
75		df-nninf 7295	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
76	74, 75	elrab2 2962	. 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)))
77	13, 70, 76	sylanbrc 417	1 ⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  ∅c0 3491  ifcif 3602   ↦ cmpt 4145  Tr wtr 4182  Ord word 4453  suc csuc 4456  ωcom 4682  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007  1_oc1o 6561  2_oc2o 6562   ↑_𝑚 cmap 6803  ℕ_∞xnninf 7294

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1o 6568  df-2o 6569  df-map 6805  df-nninf 7295

This theorem is referenced by:  nnnninf2  7302  fnn0nninf  10668  nninfinf  10673  nninfsellemdc  16406  nninfsellemqall  16411  nninffeq  16416