| 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 45271 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑍 × {𝐴})) |
| 7 | letopon 23213 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
| 8 | xlimconst.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 9 | xlimconst.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 10 | 9 | lmconst 23269 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
| 11 | 7, 4, 8, 10 | mp3an2i 1468 | . . 3 ⊢ (𝜑 → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
| 12 | 6, 11 | eqbrtrd 5165 | . 2 ⊢ (𝜑 → 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
| 13 | df-xlim 45834 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
| 14 | 13 | breqi 5149 | . 2 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
| 15 | 12, 14 | sylibr 234 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 {csn 4626 class class class wbr 5143 × cxp 5683 Fn wfn 6556 ‘cfv 6561 ℝ*cxr 11294 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 ordTopcordt 17544 TopOnctopon 22916 ⇝𝑡clm 23234 ~~>*clsxlim 45833 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-topon 22917 df-bases 22953 df-lm 23237 df-xlim 45834 |
| This theorem is referenced by: xlimconst2 45850 |
| Copyright terms: Public domain | W3C validator |