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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 22502 | . 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍)) |
3 | zfbas 22501 | . . . . 5 ⊢ ran ℤ≥ ∈ (fBas‘ℤ) | |
4 | 0nelfb 22436 | . . . . 5 ⊢ (ran ℤ≥ ∈ (fBas‘ℤ) → ¬ ∅ ∈ ran ℤ≥) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥ |
6 | imassrn 5907 | . . . . . 6 ⊢ (ℤ≥ “ 𝑍) ⊆ ran ℤ≥ | |
7 | 2, 6 | eqsstrdi 3969 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ ran ℤ≥) |
8 | 7 | sseld 3914 | . . . 4 ⊢ (𝑀 ∈ ℤ → (∅ ∈ (ran ℤ≥ ↾t 𝑍) → ∅ ∈ ran ℤ≥)) |
9 | 5, 8 | mtoi 202 | . . 3 ⊢ (𝑀 ∈ ℤ → ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍)) |
10 | uzssz 12252 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
11 | 1, 10 | eqsstri 3949 | . . . 4 ⊢ 𝑍 ⊆ ℤ |
12 | trfbas2 22448 | . . . 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 237 | . 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍)) |
15 | 2, 14 | eqeltrrd 2891 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 ran crn 5520 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ℤcz 11969 ℤ≥cuz 12231 ↾t crest 16686 fBascfbas 20079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-nn 11626 df-z 11970 df-uz 12232 df-rest 16688 df-fbas 20088 |
This theorem is referenced by: lmflf 22610 caucfil 23887 cmetcaulem 23892 |
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