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Theorem infnninf 7115
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 4669 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 6436 . . . . . 6 1o ∈ 2o
21a1i 9 . . . . 5 ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
32fmpttd 5666 . . . 4 (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
43mptru 1362 . . 3 (𝑖 ∈ ω ↦ 1o):ω⟶2o
5 2on 6419 . . . 4 2o ∈ On
6 omex 4588 . . . 4 ω ∈ V
7 elmapg 6654 . . . 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 4590 . . . . . 6 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11 eqidd 2178 . . . . . . 7 (𝑖 = suc 𝑗 → 1o = 1o)
12 eqid 2177 . . . . . . 7 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13 1oex 6418 . . . . . . 7 1o ∈ V
1411, 12, 13fvmpt 5588 . . . . . 6 (suc 𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
1510, 14syl 14 . . . . 5 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
16 eqidd 2178 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
1716, 12, 13fvmpt 5588 . . . . 5 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
1815, 17eqtr4d 2213 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19 eqimss 3209 . . . 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 2530 . 2 𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22 fveq1 5509 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23 fveq1 5509 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2422, 23sseq12d 3186 . . . 4 (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
2524ralbidv 2477 . . 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 7112 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
2725, 26elrab2 2896 . 2 ((𝑖 ∈ ω ↦ 1o) ∈ ℕ ↔ ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
289, 21, 27mpbir2an 942 1 (𝑖 ∈ ω ↦ 1o) ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wtru 1354  wcel 2148  wral 2455  Vcvv 2737  wss 3129  cmpt 4061  Oncon0 4359  suc csuc 4361  ωcom 4585  wf 5207  cfv 5211  (class class class)co 5868  1oc1o 6403  2oc2o 6404  𝑚 cmap 6641  xnninf 7111
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1o 6410  df-2o 6411  df-map 6643  df-nninf 7112
This theorem is referenced by:  nnnninf2  7118  nninfwlpoimlemdc  7168  nninffeq  14392
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