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Mirrors > Home > ILE Home > Th. List > infnninf | GIF version |
Description: The point at infinity in ℕ∞ 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.) |
Ref | Expression |
---|---|
infnninf | ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2o 6436 | . . . . . 6 ⊢ 1o ∈ 2o | |
2 | 1 | a1i 9 | . . . . 5 ⊢ ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o) |
3 | 2 | fmpttd 5666 | . . . 4 ⊢ (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o) |
4 | 3 | mptru 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) ∈ (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 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 | |
14 | 11, 12, 13 | fvmpt 5588 | . . . . . 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 2178 | . . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o) | |
17 | 16, 12, 13 | fvmpt 5588 | . . . . 5 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o) |
18 | 15, 17 | eqtr4d 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)) | |
20 | 18, 19 | syl 14 | . . 3 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)) |
21 | 20 | rgen 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) → (𝑓‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗)) | |
24 | 22, 23 | sseq12d 3186 | . . . 4 ⊢ (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))) |
25 | 24 | ralbidv 2477 | . . 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 7112 | . . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓‘𝑗)} | |
27 | 25, 26 | elrab2 2896 | . 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 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 ℕ∞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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