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 2977 | . . . 4 ⊢ Ⅎ𝑘(ℤ≥‘𝑁) | |
4 | 2, 3 | nfres 5855 | . . 3 ⊢ Ⅎ𝑘(𝐹 ↾ (ℤ≥‘𝑁)) |
5 | xlimconst2.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | xlimconst2.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
7 | 5, 6 | eluzelz2d 41707 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | eqid 2821 | . . 3 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
9 | xlimconst2.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
10 | 9 | ffnd 6515 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
11 | 5, 6 | uzssd2 41711 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
12 | 10, 11 | fnssresd 6471 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁)) Fn (ℤ≥‘𝑁)) |
13 | xlimconst2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
14 | fvres 6689 | . . . . 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 2856 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝐹 ↾ (ℤ≥‘𝑁))‘𝑘) = 𝐴) |
18 | 1, 4, 7, 8, 12, 13, 17 | xlimconst 42126 | . 2 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁))~~>*𝐴) |
19 | 5, 9 | fuzxrpmcn 42129 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
20 | 19, 7 | xlimres 42122 | . 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 2961 class class class wbr 5066 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 ℝ*cxr 10674 ℤ≥cuz 12244 ~~>*clsxlim 42119 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 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-int 4877 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-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 df-topgen 16717 df-ordt 16774 df-ps 17810 df-tsr 17811 df-top 21502 df-topon 21519 df-bases 21554 df-lm 21837 df-xlim 42120 |
This theorem is referenced by: climxlim2lem 42146 |
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