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 2979 | . . . 4 ⊢ Ⅎ𝑘(ℤ≥‘𝑁) | |
4 | 2, 3 | nfres 5857 | . . 3 ⊢ Ⅎ𝑘(𝐹 ↾ (ℤ≥‘𝑁)) |
5 | xlimconst2.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | xlimconst2.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
7 | 5, 6 | eluzelz2d 41694 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | eqid 2823 | . . 3 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
9 | xlimconst2.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
10 | 9 | ffnd 6517 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
11 | 5, 6 | uzssd2 41698 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
12 | 10, 11 | fnssresd 6473 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁)) Fn (ℤ≥‘𝑁)) |
13 | xlimconst2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
14 | fvres 6691 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → ((𝐹 ↾ (ℤ≥‘𝑁))‘𝑘) = (𝐹‘𝑘)) | |
15 | 14 | adantl 484 | . . . 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 2858 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝐹 ↾ (ℤ≥‘𝑁))‘𝑘) = 𝐴) |
18 | 1, 4, 7, 8, 12, 13, 17 | xlimconst 42113 | . 2 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁))~~>*𝐴) |
19 | 5, 9 | fuzxrpmcn 42116 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
20 | 19, 7 | xlimres 42109 | . 2 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑁))~~>*𝐴)) |
21 | 18, 20 | mpbird 259 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 class class class wbr 5068 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 ℝ*cxr 10676 ℤ≥cuz 12246 ~~>*clsxlim 42106 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-neg 10875 df-z 11985 df-uz 12247 df-topgen 16719 df-ordt 16776 df-ps 17812 df-tsr 17813 df-top 21504 df-topon 21521 df-bases 21556 df-lm 21839 df-xlim 42107 |
This theorem is referenced by: climxlim2lem 42133 |
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