Theorem infnninf 7415

Description: The point at infinity in ℕ_∞ is the constant sequence equal to 1_o. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1_o}), as fconstmpt 4797 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)

Assertion

Ref	Expression
infnninf	⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞

Proof of Theorem infnninf

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

Step	Hyp	Ref	Expression
1		1lt2o 6675	. . . . . 6 ⊢ 1o ∈ 2o
2	1	a1i 9	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
3	2	fmpttd 5832	. . . 4 ⊢ (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
4	3	mptru 1407	. . 3 ⊢ (𝑖 ∈ ω ↦ 1o):ω⟶2o
5		2on 6656	. . . 4 ⊢ 2o ∈ On
6		omex 4715	. . . 4 ⊢ ω ∈ V
7		elmapg 6895	. . . 4 ⊢ ((2o ∈ On ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o))
8	5, 6, 7	mp2an 426	. . 3 ⊢ ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o)
9	4, 8	mpbir 146	. 2 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω)
10		peano2 4717	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11		eqidd 2233	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → 1o = 1o)
12		eqid 2232	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13		1oex 6655	. . . . . . 7 ⊢ 1o ∈ V
14	11, 12, 13	fvmpt 5754	. . . . . 6 ⊢ (suc 𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
15	10, 14	syl 14	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
16		eqidd 2233	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o)
17	16, 12, 13	fvmpt 5754	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
18	15, 17	eqtr4d 2268	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19		eqimss 3292	. . . 4 ⊢ (((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗) → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
20	18, 19	syl 14	. . 3 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
21	20	rgen 2595	. 2 ⊢ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22		fveq1 5669	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23		fveq1 5669	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	22, 23	sseq12d 3269	. . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
25	24	ralbidv 2542	. . 3 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
26		df-nninf 7411	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
27	25, 26	elrab2 2976	. 2 ⊢ ((𝑖 ∈ ω ↦ 1o) ∈ ℕ∞ ↔ ((𝑖 ∈ ω ↦ 1o) ∈ (2o ↑𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
28	9, 21, 27	mpbir2an 951	1 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2203  ∀wral 2520  Vcvv 2813   ⊆ wss 3211   ↦ cmpt 4171  Oncon0 4484  suc csuc 4486  ωcom 4712  ⟶wf 5348  ‘cfv 5352  (class class class)co 6050  1_oc1o 6640  2_oc2o 6641   ↑_𝑚 cmap 6882  ℕ_∞xnninf 7410

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 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  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 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1o 6647  df-2o 6648  df-map 6884  df-nninf 7411

This theorem is referenced by:  nnnninf2  7418  nninfwlpoimlemdc  7468  nninfct  12737  nninffeq  16798  nnnninfen  16799