limsup10exlem - Mathbox for Glauco Siliprandi

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Theorem limsup10exlem 45083

Description: The range of the given function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
limsup10exlem.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))
limsup10exlem.2	⊢ (𝜑 → 𝐾 ∈ ℝ)

**Assertion**
Ref	Expression
limsup10exlem	⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})

Distinct variable groups: 𝑛,𝐾 𝜑,𝑛 Allowed substitution hint: 𝐹(𝑛)

**Proof of Theorem limsup10exlem**
Step	Hyp	Ref	Expression
1		c0ex 11230	. . . . . . 7 ⊢ 0 ∈ V
2	1	prid1 4762	. . . . . 6 ⊢ 0 ∈ {0, 1}
3		1re 11236	. . . . . . . 8 ⊢ 1 ∈ ℝ
4	3	elexi 3489	. . . . . . 7 ⊢ 1 ∈ V
5	4	prid2 4763	. . . . . 6 ⊢ 1 ∈ {0, 1}
6	2, 5	ifcli 4571	. . . . 5 ⊢ if(2 ∥ 𝑛, 0, 1) ∈ {0, 1}
7	6	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))) → if(2 ∥ 𝑛, 0, 1) ∈ {0, 1})
8	7	ralrimiva 3141	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))if(2 ∥ 𝑛, 0, 1) ∈ {0, 1})
9		nfv 1910	. . . 4 ⊢ Ⅎ𝑛𝜑
10	1, 4	ifex 4574	. . . . 5 ⊢ if(2 ∥ 𝑛, 0, 1) ∈ V
11	10	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))) → if(2 ∥ 𝑛, 0, 1) ∈ V)
12		limsup10exlem.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))
13	9, 11, 12	imassmpt 44562	. . 3 ⊢ (𝜑 → ((𝐹 “ (𝐾[,)+∞)) ⊆ {0, 1} ↔ ∀𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))if(2 ∥ 𝑛, 0, 1) ∈ {0, 1}))
14	8, 13	mpbird 257	. 2 ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) ⊆ {0, 1})
15		limsup10exlem.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
16	15	ceilcld 13832	. . . . . . . . 9 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
17		1zzd 12615	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
18	16, 17	ifcld 4570	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ)
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ)
20		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
21		2teven 16323	. . . . . . 7 ⊢ ((if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → 2 ∥ 𝑛)
22	19, 20, 21	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → 2 ∥ 𝑛)
23	22	iftrued 4532	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → if(2 ∥ 𝑛, 0, 1) = 0)
24		2nn 12307	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
26		eqid 2727	. . . . . . . 8 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
27	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ∈ ℝ)
28	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ∈ ℝ)
29	16	zred 12688	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
30	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → (⌈‘𝐾) ∈ ℝ)
31		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ 𝐾)
32	15	ceilged 13835	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
33	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ≤ (⌈‘𝐾))
34	27, 28, 30, 31, 33	letrd 11393	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ (⌈‘𝐾))
35		iftrue 4530	. . . . . . . . . . 11 ⊢ (1 ≤ 𝐾 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = (⌈‘𝐾))
36	35	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = (⌈‘𝐾))
37	34, 36	breqtrrd 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
38	3	leidi 11770	. . . . . . . . . . 11 ⊢ 1 ≤ 1
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ 1)
40		iffalse 4533	. . . . . . . . . . 11 ⊢ (¬ 1 ≤ 𝐾 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = 1)
41	40	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = 1)
42	39, 41	breqtrrd 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
43	37, 42	pm2.61dan 812	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
44	26, 17, 18, 43	eluzd 44714	. . . . . . 7 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ (ℤ_≥‘1))
45		nnuz 12887	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
46	44, 45	eleqtrrdi 2839	. . . . . 6 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℕ)
47	25, 46	nnmulcld 12287	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℕ)
48	1	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ V)
49	12, 23, 47, 48	fvmptd2 7007	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) = 0)
50	10, 12	fnmpti 6692	. . . . . 6 ⊢ 𝐹 Fn ℕ
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 Fn ℕ)
52	15	rexrd 11286	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
53		pnfxr 11290	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
55	47	nnxrd 44578	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ^*)
56	47	nnred 12249	. . . . . . 7 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ)
57	46	nnred 12249	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ)
58	33, 36	breqtrrd 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
59	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ∈ ℝ)
60	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ∈ ℝ)
61		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → ¬ 1 ≤ 𝐾)
62	59, 60, 61	nleltd 44757	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 < 1)
63	59, 60, 62	ltled 11384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ 1)
64	41	eqcomd 2733	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 = if(1 ≤ 𝐾, (⌈‘𝐾), 1))
65	63, 64	breqtrd 5168	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
66	58, 65	pm2.61dan 812	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
67	46	nnrpd 13038	. . . . . . . . 9 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ⁺)
68		2timesgt 44593	. . . . . . . . 9 ⊢ (if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ⁺ → if(1 ≤ 𝐾, (⌈‘𝐾), 1) < (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
69	67, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) < (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
70	15, 57, 56, 66, 69	lelttrd 11394	. . . . . . 7 ⊢ (𝜑 → 𝐾 < (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
71	15, 56, 70	ltled 11384	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
72	56	ltpnfd 13125	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < +∞)
73	52, 54, 55, 71, 72	elicod 13398	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ (𝐾[,)+∞))
74	51, 47, 73	fnfvimad 7240	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) ∈ (𝐹 “ (𝐾[,)+∞)))
75	49, 74	eqeltrrd 2829	. . 3 ⊢ (𝜑 → 0 ∈ (𝐹 “ (𝐾[,)+∞)))
76	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ)
77		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
78		2tp1odd 16320	. . . . . . 7 ⊢ ((if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → ¬ 2 ∥ 𝑛)
79	76, 77, 78	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → ¬ 2 ∥ 𝑛)
80	79	iffalsed 4535	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → if(2 ∥ 𝑛, 0, 1) = 1)
81	47	peano2nnd 12251	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℕ)
82		1xr 11295	. . . . . 6 ⊢ 1 ∈ ℝ^*
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ^*)
84	12, 80, 81, 83	fvmptd2 7007	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) = 1)
85	81	nnxrd 44578	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ^*)
86	81	nnred 12249	. . . . . . 7 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ)
87	56	ltp1d 12166	. . . . . . . 8 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
88	15, 56, 86, 70, 87	lttrd 11397	. . . . . . 7 ⊢ (𝜑 → 𝐾 < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
89	15, 86, 88	ltled 11384	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
90	86	ltpnfd 13125	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) < +∞)
91	52, 54, 85, 89, 90	elicod 13398	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ (𝐾[,)+∞))
92	51, 81, 91	fnfvimad 7240	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) ∈ (𝐹 “ (𝐾[,)+∞)))
93	84, 92	eqeltrrd 2829	. . 3 ⊢ (𝜑 → 1 ∈ (𝐹 “ (𝐾[,)+∞)))
94	75, 93	prssd 4821	. 2 ⊢ (𝜑 → {0, 1} ⊆ (𝐹 “ (𝐾[,)+∞)))
95	14, 94	eqssd 3995	1 ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ∩ cin 3943 ⊆ wss 3944 ifcif 4524 {cpr 4626 class class class wbr 5142 ↦ cmpt 5225 “ cima 5675 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 ℕcn 12234 2c2 12289 ℤcz 12580 ℤ_≥cuz 12844 ℝ⁺crp 12998 [,)cico 13350 ⌈cceil 13780 ∥ cdvds 16222

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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-ico 13354 df-fl 13781 df-ceil 13782 df-dvds 16223

This theorem is referenced by: limsup10ex 45084 liminf10ex 45085

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