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Mathbox for Glauco Siliprandi |
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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 45209 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑍 × {𝐴})) |
7 | letopon 23228 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
8 | xlimconst.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
9 | xlimconst.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
10 | 9 | lmconst 23284 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
11 | 7, 4, 8, 10 | mp3an2i 1465 | . . 3 ⊢ (𝜑 → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
12 | 6, 11 | eqbrtrd 5169 | . 2 ⊢ (𝜑 → 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
13 | df-xlim 45774 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
14 | 13 | breqi 5153 | . 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 1536 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 {csn 4630 class class class wbr 5147 × cxp 5686 Fn wfn 6557 ‘cfv 6562 ℝ*cxr 11291 ≤ cle 11293 ℤcz 12610 ℤ≥cuz 12875 ordTopcordt 17545 TopOnctopon 22931 ⇝𝑡clm 23249 ~~>*clsxlim 45773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-1o 8504 df-2o 8505 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-neg 11492 df-z 12611 df-uz 12876 df-topgen 17489 df-ordt 17547 df-ps 18623 df-tsr 18624 df-top 22915 df-topon 22932 df-bases 22968 df-lm 23252 df-xlim 45774 |
This theorem is referenced by: xlimconst2 45790 |
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