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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 40284 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑍 × {𝐴})) |
7 | letopon 21380 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
8 | xlimconst.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
9 | xlimconst.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
10 | 9 | lmconst 21436 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
11 | 7, 4, 8, 10 | mp3an2i 1596 | . . 3 ⊢ (𝜑 → (𝑍 × {𝐴})(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
12 | 6, 11 | eqbrtrd 4895 | . 2 ⊢ (𝜑 → 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
13 | df-xlim 40840 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
14 | 13 | breqi 4879 | . 2 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
15 | 12, 14 | sylibr 226 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 Ⅎwnfc 2956 {csn 4397 class class class wbr 4873 × cxp 5340 Fn wfn 6118 ‘cfv 6123 ℝ*cxr 10390 ≤ cle 10392 ℤcz 11704 ℤ≥cuz 11968 ordTopcordt 16512 TopOnctopon 21085 ⇝𝑡clm 21401 ~~>*clsxlim 40839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-neg 10588 df-z 11705 df-uz 11969 df-topgen 16457 df-ordt 16514 df-ps 17553 df-tsr 17554 df-top 21069 df-topon 21086 df-bases 21121 df-lm 21404 df-xlim 40840 |
This theorem is referenced by: xlimconst2 40856 |
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