xlimcl - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimcl		Structured version Visualization version GIF version

Theorem xlimcl 44538

Description: The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Assertion**
Ref	Expression
xlimcl	⊢ (𝐹~~>𝐴 → 𝐴 ∈ ℝ^)

**Proof of Theorem xlimcl**
Step	Hyp	Ref	Expression
1		letopon 22709	. 2 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2		df-xlim 44535	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
3	2	breqi 5155	. . 3 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴)
4	3	biimpi 215	. 2 ⊢ (𝐹~~>*𝐴 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴)
5		lmcl 22801	. 2 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴) → 𝐴 ∈ ℝ^)
6	1, 4, 5	sylancr 588	1 ⊢ (𝐹~~>𝐴 → 𝐴 ∈ ℝ^)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 ℝ^*cxr 11247 ≤ cle 11249 ordTopcordt 17445 TopOnctopon 22412 ⇝_𝑡clm 22730 ~~>*clsxlim 44534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-topgen 17389 df-ordt 17447 df-ps 18519 df-tsr 18520 df-top 22396 df-topon 22413 df-bases 22449 df-lm 22733 df-xlim 44535

This theorem is referenced by: dfxlim2v 44563 xlimliminflimsup 44578

W3C validator