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Mirrors > Home > MPE Home > Th. List > uzfbas | Structured version Visualization version GIF version |
Description: The set of upper sets of integers based at a point in a fixed upper integer set like ℕ is a filter base on ℕ, which corresponds to convergence of sequences on ℕ. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
uzfbas.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
uzfbas | ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzfbas.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | uzrest 23271 | . 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍)) |
3 | zfbas 23270 | . . . . 5 ⊢ ran ℤ≥ ∈ (fBas‘ℤ) | |
4 | 0nelfb 23205 | . . . . 5 ⊢ (ran ℤ≥ ∈ (fBas‘ℤ) → ¬ ∅ ∈ ran ℤ≥) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥ |
6 | imassrn 6028 | . . . . . 6 ⊢ (ℤ≥ “ 𝑍) ⊆ ran ℤ≥ | |
7 | 2, 6 | eqsstrdi 4002 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ ran ℤ≥) |
8 | 7 | sseld 3947 | . . . 4 ⊢ (𝑀 ∈ ℤ → (∅ ∈ (ran ℤ≥ ↾t 𝑍) → ∅ ∈ ran ℤ≥)) |
9 | 5, 8 | mtoi 198 | . . 3 ⊢ (𝑀 ∈ ℤ → ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍)) |
10 | uzssz 12792 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
11 | 1, 10 | eqsstri 3982 | . . . 4 ⊢ 𝑍 ⊆ ℤ |
12 | trfbas2 23217 | . . . 4 ⊢ ((ran ℤ≥ ∈ (fBas‘ℤ) ∧ 𝑍 ⊆ ℤ) → ((ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍) ↔ ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍))) | |
13 | 3, 11, 12 | mp2an 691 | . . 3 ⊢ ((ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍) ↔ ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍)) |
14 | 9, 13 | sylibr 233 | . 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍)) |
15 | 2, 14 | eqeltrrd 2835 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 ∅c0 4286 ran crn 5638 “ cima 5640 ‘cfv 6500 (class class class)co 7361 ℤcz 12507 ℤ≥cuz 12771 ↾t crest 17310 fBascfbas 20807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-i2m1 11127 ax-1ne0 11128 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-neg 11396 df-nn 12162 df-z 12508 df-uz 12772 df-rest 17312 df-fbas 20816 |
This theorem is referenced by: lmflf 23379 caucfil 24670 cmetcaulem 24675 |
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