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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 22505 | . 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍)) |
3 | zfbas 22504 | . . . . 5 ⊢ ran ℤ≥ ∈ (fBas‘ℤ) | |
4 | 0nelfb 22439 | . . . . 5 ⊢ (ran ℤ≥ ∈ (fBas‘ℤ) → ¬ ∅ ∈ ran ℤ≥) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ¬ ∅ ∈ ran ℤ≥ |
6 | imassrn 5940 | . . . . . 6 ⊢ (ℤ≥ “ 𝑍) ⊆ ran ℤ≥ | |
7 | 2, 6 | eqsstrdi 4021 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ ran ℤ≥) |
8 | 7 | sseld 3966 | . . . 4 ⊢ (𝑀 ∈ ℤ → (∅ ∈ (ran ℤ≥ ↾t 𝑍) → ∅ ∈ ran ℤ≥)) |
9 | 5, 8 | mtoi 201 | . . 3 ⊢ (𝑀 ∈ ℤ → ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍)) |
10 | uzssz 12265 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
11 | 1, 10 | eqsstri 4001 | . . . 4 ⊢ 𝑍 ⊆ ℤ |
12 | trfbas2 22451 | . . . 4 ⊢ ((ran ℤ≥ ∈ (fBas‘ℤ) ∧ 𝑍 ⊆ ℤ) → ((ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍) ↔ ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍))) | |
13 | 3, 11, 12 | mp2an 690 | . . 3 ⊢ ((ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍) ↔ ¬ ∅ ∈ (ran ℤ≥ ↾t 𝑍)) |
14 | 9, 13 | sylibr 236 | . 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ∈ (fBas‘𝑍)) |
15 | 2, 14 | eqeltrrd 2914 | 1 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 ran crn 5556 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ℤcz 11982 ℤ≥cuz 12244 ↾t crest 16694 fBascfbas 20533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-i2m1 10605 ax-1ne0 10606 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-nn 11639 df-z 11983 df-uz 12245 df-rest 16696 df-fbas 20542 |
This theorem is referenced by: lmflf 22613 caucfil 23886 cmetcaulem 23891 |
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