Theorem infnninf 7190

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 4710 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 6500	. . . . . 6 ⊢ 1o ∈ 2o
2	1	a1i 9	. . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
3	2	fmpttd 5717	. . . 4 ⊢ (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
4	3	mptru 1373	. . 3 ⊢ (𝑖 ∈ ω ↦ 1o):ω⟶2o
5		2on 6483	. . . 4 ⊢ 2o ∈ On
6		omex 4629	. . . 4 ⊢ ω ∈ V
7		elmapg 6720	. . . 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 4631	. . . . . 6 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11		eqidd 2197	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → 1o = 1o)
12		eqid 2196	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13		1oex 6482	. . . . . . 7 ⊢ 1o ∈ V
14	11, 12, 13	fvmpt 5638	. . . . . 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 2197	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o)
17	16, 12, 13	fvmpt 5638	. . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
18	15, 17	eqtr4d 2232	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19		eqimss 3237	. . . 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 2550	. 2 ⊢ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22		fveq1 5557	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23		fveq1 5557	. . . . 5 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
24	22, 23	sseq12d 3214	. . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
25	24	ralbidv 2497	. . 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 7186	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)}
27	25, 26	elrab2 2923	. 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 944	1 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ⊤wtru 1365   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157   ↦ cmpt 4094  Oncon0 4398  suc csuc 4400  ωcom 4626  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  1_oc1o 6467  2_oc2o 6468   ↑_𝑚 cmap 6707  ℕ_∞xnninf 7185

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1o 6474  df-2o 6475  df-map 6709  df-nninf 7186

This theorem is referenced by:  nnnninf2  7193  nninfwlpoimlemdc  7243  nninfct  12208  nninffeq  15664  nnnninfen  15665