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Mirrors > Home > ILE Home > Th. List > inftonninf | GIF version |
Description: The mapping of +∞ into ℕ∞ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.) |
Ref | Expression |
---|---|
fxnn0nninf.g | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
fxnn0nninf.f | ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) |
fxnn0nninf.i | ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) |
Ref | Expression |
---|---|
inftonninf | ⊢ (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fxnn0nninf.i | . . 3 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) | |
2 | 1 | fveq1i 5422 | . 2 ⊢ (𝐼‘+∞) = (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘+∞) |
3 | pnf0xnn0 9047 | . . 3 ⊢ +∞ ∈ ℕ0* | |
4 | omex 4507 | . . . 4 ⊢ ω ∈ V | |
5 | 1oex 6321 | . . . . 5 ⊢ 1o ∈ V | |
6 | 5 | snex 4109 | . . . 4 ⊢ {1o} ∈ V |
7 | 4, 6 | xpex 4654 | . . 3 ⊢ (ω × {1o}) ∈ V |
8 | pnfnre 7807 | . . . . . 6 ⊢ +∞ ∉ ℝ | |
9 | 8 | neli 2405 | . . . . 5 ⊢ ¬ +∞ ∈ ℝ |
10 | nn0re 8986 | . . . . 5 ⊢ (+∞ ∈ ℕ0 → +∞ ∈ ℝ) | |
11 | 9, 10 | mto 651 | . . . 4 ⊢ ¬ +∞ ∈ ℕ0 |
12 | fxnn0nninf.g | . . . . . . 7 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
13 | fxnn0nninf.f | . . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) | |
14 | 12, 13 | fnn0nninf 10210 | . . . . . 6 ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ |
15 | 14 | fdmi 5280 | . . . . 5 ⊢ dom (𝐹 ∘ ◡𝐺) = ℕ0 |
16 | 15 | eleq2i 2206 | . . . 4 ⊢ (+∞ ∈ dom (𝐹 ∘ ◡𝐺) ↔ +∞ ∈ ℕ0) |
17 | 11, 16 | mtbir 660 | . . 3 ⊢ ¬ +∞ ∈ dom (𝐹 ∘ ◡𝐺) |
18 | fsnunfv 5621 | . . 3 ⊢ ((+∞ ∈ ℕ0* ∧ (ω × {1o}) ∈ V ∧ ¬ +∞ ∈ dom (𝐹 ∘ ◡𝐺)) → (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘+∞) = (ω × {1o})) | |
19 | 3, 7, 17, 18 | mp3an 1315 | . 2 ⊢ (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘+∞) = (ω × {1o}) |
20 | fconstmpt 4586 | . 2 ⊢ (ω × {1o}) = (𝑥 ∈ ω ↦ 1o) | |
21 | 2, 19, 20 | 3eqtri 2164 | 1 ⊢ (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ∅c0 3363 ifcif 3474 {csn 3527 〈cop 3530 ↦ cmpt 3989 ωcom 4504 × cxp 4537 ◡ccnv 4538 dom cdm 4539 ∘ ccom 4543 ‘cfv 5123 (class class class)co 5774 freccfrec 6287 1oc1o 6306 ℕ∞xnninf 7005 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 +∞cpnf 7797 ℕ0cn0 8977 ℕ0*cxnn0 9040 ℤcz 9054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-1o 6313 df-2o 6314 df-map 6544 df-nninf 7007 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-xnn0 9041 df-z 9055 df-uz 9327 |
This theorem is referenced by: (None) |
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