limsup10ex - Mathbox for Glauco Siliprandi

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Theorem limsup10ex 45161

Description: The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypothesis**
Ref	Expression
limsup10ex.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))

**Assertion**
Ref	Expression
limsup10ex	⊢ (lim sup‘𝐹) = 1

Proof of Theorem limsup10ex Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nftru 1799	. . . 4 ⊢ Ⅎ𝑘⊤
2		nnex 12248	. . . . 5 ⊢ ℕ ∈ V
3	2	a1i 11	. . . 4 ⊢ (⊤ → ℕ ∈ V)
4		limsup10ex.1	. . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))
5		0xr 11291	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
6	5	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ^*)
7		1xr 11303	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
8	7	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ^*)
9	6, 8	ifcld 4575	. . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ^*)
10	4, 9	fmpti 7122	. . . . 5 ⊢ 𝐹:ℕ⟶ℝ^*
11	10	a1i 11	. . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ^*)
12		eqid 2728	. . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^, < )) = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^, < ))
13	1, 3, 11, 12	limsupval3 45080	. . 3 ⊢ (⊤ → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^, < )), ℝ^, < ))
14	13	mptru 1541	. 2 ⊢ (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^, < )), ℝ^, < )
15		id 22	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ)
16	4, 15	limsup10exlem 45160	. . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1})
17	16	supeq1d 9469	. . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ^, < ) = sup({0, 1}, ℝ^, < ))
18		xrltso 13152	. . . . . . . . 9 ⊢ < Or ℝ^*
19		suppr 9494	. . . . . . . . 9 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^* ∧ 1 ∈ ℝ^) → sup({0, 1}, ℝ^, < ) = if(1 < 0, 0, 1))
20	18, 5, 7, 19	mp3an 1458	. . . . . . . 8 ⊢ sup({0, 1}, ℝ^*, < ) = if(1 < 0, 0, 1)
21		0le1 11767	. . . . . . . . . 10 ⊢ 0 ≤ 1
22		0re 11246	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
23		1re 11244	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
24	22, 23	lenlti 11364	. . . . . . . . . 10 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0)
25	21, 24	mpbi 229	. . . . . . . . 9 ⊢ ¬ 1 < 0
26	25	iffalsei 4539	. . . . . . . 8 ⊢ if(1 < 0, 0, 1) = 1
27	20, 26	eqtri 2756	. . . . . . 7 ⊢ sup({0, 1}, ℝ^*, < ) = 1
28	17, 27	eqtrdi 2784	. . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ^*, < ) = 1)
29	28	mpteq2ia 5251	. . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^*, < )) = (𝑘 ∈ ℝ ↦ 1)
30	29	rneqi 5939	. . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^*, < )) = ran (𝑘 ∈ ℝ ↦ 1)
31		eqid 2728	. . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 1) = (𝑘 ∈ ℝ ↦ 1)
32		ren0 44784	. . . . . . 7 ⊢ ℝ ≠ ∅
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ≠ ∅)
34	31, 33	rnmptc 7219	. . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 1) = {1})
35	34	mptru 1541	. . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 1) = {1}
36	30, 35	eqtri 2756	. . 3 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^*, < )) = {1}
37	36	infeq1i 9501	. 2 ⊢ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ^, < )), ℝ^, < ) = inf({1}, ℝ^*, < )
38		infsn 9528	. . 3 ⊢ (( < Or ℝ^* ∧ 1 ∈ ℝ^) → inf({1}, ℝ^, < ) = 1)
39	18, 7, 38	mp2an 691	. 2 ⊢ inf({1}, ℝ^*, < ) = 1
40	14, 37, 39	3eqtri 2760	1 ⊢ (lim sup‘𝐹) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∅c0 4323 ifcif 4529 {csn 4629 {cpr 4631 class class class wbr 5148 ↦ cmpt 5231 Or wor 5589 ran crn 5679 “ cima 5681 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 supcsup 9463 infcinf 9464 ℝcr 11137 0cc0 11138 1c1 11139 +∞cpnf 11275 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 ℕcn 12242 2c2 12297 [,)cico 13358 lim supclsp 15446 ∥ cdvds 16230

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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

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-rmo 3373 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-iun 4998 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-pred 6305 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-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-fl 13789 df-ceil 13790 df-limsup 15447 df-dvds 16231

This theorem is referenced by: liminfltlimsupex 45169

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