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Mirrors > Home > ILE Home > Th. List > fxnn0nninf | GIF version |
Description: A function from ℕ0* into ℕ∞. (Contributed by Jim Kingdon, 16-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 |
---|---|
fxnn0nninf | ⊢ 𝐼:ℕ0*⟶ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fxnn0nninf.g | . . . . . 6 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
2 | fxnn0nninf.f | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) | |
3 | 1, 2 | fnn0nninf 10210 | . . . . 5 ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ |
4 | pnfex 7819 | . . . . . . . 8 ⊢ +∞ ∈ V | |
5 | omex 4507 | . . . . . . . . 9 ⊢ ω ∈ V | |
6 | 1oex 6321 | . . . . . . . . . 10 ⊢ 1o ∈ V | |
7 | 6 | snex 4109 | . . . . . . . . 9 ⊢ {1o} ∈ V |
8 | 5, 7 | xpex 4654 | . . . . . . . 8 ⊢ (ω × {1o}) ∈ V |
9 | 4, 8 | f1osn 5407 | . . . . . . 7 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}–1-1-onto→{(ω × {1o})} |
10 | f1of 5367 | . . . . . . 7 ⊢ ({〈+∞, (ω × {1o})〉}:{+∞}–1-1-onto→{(ω × {1o})} → {〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})}) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})} |
12 | infnninf 7022 | . . . . . . 7 ⊢ (ω × {1o}) ∈ ℕ∞ | |
13 | snssi 3664 | . . . . . . 7 ⊢ ((ω × {1o}) ∈ ℕ∞ → {(ω × {1o})} ⊆ ℕ∞) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ {(ω × {1o})} ⊆ ℕ∞ |
15 | fss 5284 | . . . . . 6 ⊢ (({〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})} ∧ {(ω × {1o})} ⊆ ℕ∞) → {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) | |
16 | 11, 14, 15 | mp2an 422 | . . . . 5 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞ |
17 | 3, 16 | pm3.2i 270 | . . . 4 ⊢ ((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) |
18 | disj 3411 | . . . . 5 ⊢ ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞}) | |
19 | nn0nepnf 9048 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ≠ +∞) | |
20 | 19 | neneqd 2329 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞) |
21 | elsni 3545 | . . . . . 6 ⊢ (𝑥 ∈ {+∞} → 𝑥 = +∞) | |
22 | 20, 21 | nsyl 617 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞}) |
23 | 18, 22 | mprgbir 2490 | . . . 4 ⊢ (ℕ0 ∩ {+∞}) = ∅ |
24 | fun2 5296 | . . . 4 ⊢ ((((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) ∧ (ℕ0 ∩ {+∞}) = ∅) → ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞) | |
25 | 17, 23, 24 | mp2an 422 | . . 3 ⊢ ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞ |
26 | fxnn0nninf.i | . . . 4 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) | |
27 | 26 | feq1i 5265 | . . 3 ⊢ (𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ ↔ ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞) |
28 | 25, 27 | mpbir 145 | . 2 ⊢ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ |
29 | df-xnn0 9041 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
30 | 29 | feq2i 5266 | . 2 ⊢ (𝐼:ℕ0*⟶ℕ∞ ↔ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞) |
31 | 28, 30 | mpbir 145 | 1 ⊢ 𝐼:ℕ0*⟶ℕ∞ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∪ cun 3069 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 ifcif 3474 {csn 3527 〈cop 3530 ↦ cmpt 3989 ωcom 4504 × cxp 4537 ◡ccnv 4538 ∘ ccom 4543 ⟶wf 5119 –1-1-onto→wf1o 5122 (class class class)co 5774 freccfrec 6287 1oc1o 6306 ℕ∞xnninf 7005 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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