![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimconst | Structured version Visualization version GIF version |
Description: A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimconst.p | ⊢ Ⅎ𝑘𝜑 |
xlimconst.k | ⊢ Ⅎ𝑘𝐹 |
xlimconst.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimconst.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimconst.f | ⊢ (𝜑 → 𝐹 Fn 𝑍) |
xlimconst.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xlimconst.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
Ref | Expression |
---|---|
xlimconst | ⊢ (𝜑 → 𝐹~~>*𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimconst.p | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
2 | xlimconst.k | . . . 4 ⊢ Ⅎ𝑘𝐹 | |
3 | xlimconst.f | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑍) | |
4 | xlimconst.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | xlimconst.e | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
6 | 1, 2, 3, 4, 5 | fconst7 44613 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑍 × {𝐴})) |
7 | letopon 23102 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
8 | xlimconst.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
9 | xlimconst.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
10 | 9 | lmconst 23158 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
11 | 7, 4, 8, 10 | mp3an2i 1463 | . . 3 ⊢ (𝜑 → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
12 | 6, 11 | eqbrtrd 5164 | . 2 ⊢ (𝜑 → 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
13 | df-xlim 45179 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
14 | 13 | breqi 5148 | . 2 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
15 | 12, 14 | sylibr 233 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2878 {csn 4624 class class class wbr 5142 × cxp 5670 Fn wfn 6537 ‘cfv 6542 ℝ*cxr 11271 ≤ cle 11273 ℤcz 12582 ℤ≥cuz 12846 ordTopcordt 17474 TopOnctopon 22805 ⇝𝑡clm 23123 ~~>*clsxlim 45178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-1o 8480 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9428 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-neg 11471 df-z 12583 df-uz 12847 df-topgen 17418 df-ordt 17476 df-ps 18551 df-tsr 18552 df-top 22789 df-topon 22806 df-bases 22842 df-lm 23126 df-xlim 45179 |
This theorem is referenced by: xlimconst2 45195 |
Copyright terms: Public domain | W3C validator |