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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimcl | Structured version Visualization version GIF version |
Description: The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimcl | ⊢ (𝐹~~>*𝐴 → 𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letopon 22552 | . 2 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
2 | df-xlim 44030 | . . . 4 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
3 | 2 | breqi 5110 | . . 3 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
4 | 3 | biimpi 215 | . 2 ⊢ (𝐹~~>*𝐴 → 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
5 | lmcl 22644 | . 2 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) → 𝐴 ∈ ℝ*) | |
6 | 1, 4, 5 | sylancr 587 | 1 ⊢ (𝐹~~>*𝐴 → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5104 ‘cfv 6494 ℝ*cxr 11185 ≤ cle 11187 ordTopcordt 17378 TopOnctopon 22255 ⇝𝑡clm 22573 ~~>*clsxlim 44029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-pre-lttri 11122 ax-pre-lttrn 11123 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-om 7800 df-1o 8409 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fi 9344 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-topgen 17322 df-ordt 17380 df-ps 18452 df-tsr 18453 df-top 22239 df-topon 22256 df-bases 22292 df-lm 22576 df-xlim 44030 |
This theorem is referenced by: dfxlim2v 44058 xlimliminflimsup 44073 |
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