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Theorem infnninf 7323
Description: The point at infinity in is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1o}), as fconstmpt 4773 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.
StepHypRef Expression
1 1lt2o 6610 . . . . . 6 1o ∈ 2o
21a1i 9 . . . . 5 ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
32fmpttd 5802 . . . 4 (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
43mptru 1406 . . 3 (𝑖 ∈ ω ↦ 1o):ω⟶2o
5 2on 6591 . . . 4 2o ∈ On
6 omex 4691 . . . 4 ω ∈ V
7 elmapg 6830 . . . 4 ((2o ∈ On ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o))
85, 6, 7mp2an 426 . . 3 ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o)
94, 8mpbir 146 . 2 (𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω)
10 peano2 4693 . . . . . 6 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11 eqidd 2232 . . . . . . 7 (𝑖 = suc 𝑗 → 1o = 1o)
12 eqid 2231 . . . . . . 7 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13 1oex 6590 . . . . . . 7 1o ∈ V
1411, 12, 13fvmpt 5723 . . . . . 6 (suc 𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
1510, 14syl 14 . . . . 5 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
16 eqidd 2232 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
1716, 12, 13fvmpt 5723 . . . . 5 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
1815, 17eqtr4d 2267 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19 eqimss 3281 . . . 4 (((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗) → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2018, 19syl 14 . . 3 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2120rgen 2585 . 2 𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22 fveq1 5638 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23 fveq1 5638 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2422, 23sseq12d 3258 . . . 4 (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
2524ralbidv 2532 . . 3 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
26 df-nninf 7319 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
2725, 26elrab2 2965 . 2 ((𝑖 ∈ ω ↦ 1o) ∈ ℕ ↔ ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
289, 21, 27mpbir2an 950 1 (𝑖 ∈ ω ↦ 1o) ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1397  wtru 1398  wcel 2202  wral 2510  Vcvv 2802  wss 3200  cmpt 4150  Oncon0 4460  suc csuc 4462  ωcom 4688  wf 5322  cfv 5326  (class class class)co 6018  1oc1o 6575  2oc2o 6576  𝑚 cmap 6817  xnninf 7318
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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1o 6582  df-2o 6583  df-map 6819  df-nninf 7319
This theorem is referenced by:  nnnninf2  7326  nninfwlpoimlemdc  7376  nninfct  12614  nninffeq  16643  nnnninfen  16644
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