Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimconst2 | Structured version Visualization version GIF version |
Description: A sequence that eventually becomes constant, converges to its constant value (w.r.t. the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimconst2.p | ⊢ Ⅎ𝑘𝜑 |
xlimconst2.k | ⊢ Ⅎ𝑘𝐹 |
xlimconst2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimconst2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
xlimconst2.n | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
xlimconst2.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xlimconst2.e | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) = 𝐴) |
Ref | Expression |
---|---|
xlimconst2 | ⊢ (𝜑 → 𝐹~~>*𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimconst2.p | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | xlimconst2.k | . . . 4 ⊢ Ⅎ𝑘𝐹 | |
3 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑘(ℤ≥‘𝑁) | |
4 | 2, 3 | nfres 5853 | . . 3 ⊢ Ⅎ𝑘(𝐹 ↾ (ℤ≥‘𝑁)) |
5 | xlimconst2.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | xlimconst2.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
7 | 5, 6 | eluzelz2d 42626 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | eqid 2737 | . . 3 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
9 | xlimconst2.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
10 | 9 | ffnd 6546 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
11 | 5, 6 | uzssd2 42630 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
12 | 10, 11 | fnssresd 6501 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁)) Fn (ℤ≥‘𝑁)) |
13 | xlimconst2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
14 | fvres 6736 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → ((𝐹 ↾ (ℤ≥‘𝑁))‘𝑘) = (𝐹‘𝑘)) | |
15 | 14 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝐹 ↾ (ℤ≥‘𝑁))‘𝑘) = (𝐹‘𝑘)) |
16 | xlimconst2.e | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) = 𝐴) | |
17 | 15, 16 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝐹 ↾ (ℤ≥‘𝑁))‘𝑘) = 𝐴) |
18 | 1, 4, 7, 8, 12, 13, 17 | xlimconst 43041 | . 2 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁))~~>*𝐴) |
19 | 5, 9 | fuzxrpmcn 43044 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
20 | 19, 7 | xlimres 43037 | . 2 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑁))~~>*𝐴)) |
21 | 18, 20 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 class class class wbr 5053 ↾ cres 5553 ⟶wf 6376 ‘cfv 6380 ℝ*cxr 10866 ℤ≥cuz 12438 ~~>*clsxlim 43034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-1o 8202 df-er 8391 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-neg 11065 df-z 12177 df-uz 12439 df-topgen 16948 df-ordt 17006 df-ps 18072 df-tsr 18073 df-top 21791 df-topon 21808 df-bases 21843 df-lm 22126 df-xlim 43035 |
This theorem is referenced by: climxlim2lem 43061 |
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