xlimcl - Mathbox for Glauco Siliprandi

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Theorem xlimcl 45084

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 23053	. 2 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2		df-xlim 45081	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
3	2	breqi 5145	. . 3 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴)
4	3	biimpi 215	. 2 ⊢ (𝐹~~>*𝐴 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴)
5		lmcl 23145	. 2 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴) → 𝐴 ∈ ℝ^)
6	1, 4, 5	sylancr 586	1 ⊢ (𝐹~~>𝐴 → 𝐴 ∈ ℝ^)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5139 ‘cfv 6534 ℝ^*cxr 11246 ≤ cle 11248 ordTopcordt 17450 TopOnctopon 22756 ⇝_𝑡clm 23074 ~~>*clsxlim 45080

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-pre-lttri 11181 ax-pre-lttrn 11182

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-topgen 17394 df-ordt 17452 df-ps 18527 df-tsr 18528 df-top 22740 df-topon 22757 df-bases 22793 df-lm 23077 df-xlim 45081

This theorem is referenced by: dfxlim2v 45109 xlimliminflimsup 45124

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