xlimcl - Mathbox for Glauco Siliprandi

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Theorem xlimcl 45210

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 23122	. 2 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2		df-xlim 45207	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
3	2	breqi 5154	. . 3 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴)
4	3	biimpi 215	. 2 ⊢ (𝐹~~>*𝐴 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))𝐴)
5		lmcl 23214	. 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 2099 class class class wbr 5148 ‘cfv 6548 ℝ^*cxr 11278 ≤ cle 11280 ordTopcordt 17481 TopOnctopon 22825 ⇝_𝑡clm 23143 ~~>*clsxlim 45206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9435 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-topgen 17425 df-ordt 17483 df-ps 18558 df-tsr 18559 df-top 22809 df-topon 22826 df-bases 22862 df-lm 23146 df-xlim 45207

This theorem is referenced by: dfxlim2v 45235 xlimliminflimsup 45250

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