limsup10exlem - Mathbox for Glauco Siliprandi

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Theorem limsup10exlem 45693

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 11284	. . . . . . 7 ⊢ 0 ∈ V
2	1	prid1 4787	. . . . . 6 ⊢ 0 ∈ {0, 1}
3		1re 11290	. . . . . . . 8 ⊢ 1 ∈ ℝ
4	3	elexi 3511	. . . . . . 7 ⊢ 1 ∈ V
5	4	prid2 4788	. . . . . 6 ⊢ 1 ∈ {0, 1}
6	2, 5	ifcli 4595	. . . . 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 3152	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))if(2 ∥ 𝑛, 0, 1) ∈ {0, 1})
9		nfv 1913	. . . 4 ⊢ Ⅎ𝑛𝜑
10	1, 4	ifex 4598	. . . . 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 45172	. . 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 13894	. . . . . . . . 9 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
17		1zzd 12674	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
18	16, 17	ifcld 4594	. . . . . . . 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 16403	. . . . . . 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 4556	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → if(2 ∥ 𝑛, 0, 1) = 0)
24		2nn 12366	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
26		eqid 2740	. . . . . . . 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 12747	. . . . . . . . . . . 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 13897	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
33	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ≤ (⌈‘𝐾))
34	27, 28, 30, 31, 33	letrd 11447	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ (⌈‘𝐾))
35		iftrue 4554	. . . . . . . . . . 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 5194	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
38	3	leidi 11824	. . . . . . . . . . 11 ⊢ 1 ≤ 1
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ 1)
40		iffalse 4557	. . . . . . . . . . 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 5194	. . . . . . . . 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 45324	. . . . . . 7 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ (ℤ_≥‘1))
45		nnuz 12946	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
46	44, 45	eleqtrrdi 2855	. . . . . 6 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℕ)
47	25, 46	nnmulcld 12346	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℕ)
48	1	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ V)
49	12, 23, 47, 48	fvmptd2 7037	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) = 0)
50	10, 12	fnmpti 6723	. . . . . 6 ⊢ 𝐹 Fn ℕ
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 Fn ℕ)
52	15	rexrd 11340	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
53		pnfxr 11344	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
55	47	nnxrd 45188	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ^*)
56	47	nnred 12308	. . . . . . 7 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ)
57	46	nnred 12308	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ)
58	33, 36	breqtrrd 5194	. . . . . . . . 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 45367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 < 1)
63	59, 60, 62	ltled 11438	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ 1)
64	41	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 = if(1 ≤ 𝐾, (⌈‘𝐾), 1))
65	63, 64	breqtrd 5192	. . . . . . . . 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 13097	. . . . . . . . 9 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ⁺)
68		2timesgt 45203	. . . . . . . . 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 11448	. . . . . . 7 ⊢ (𝜑 → 𝐾 < (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
71	15, 56, 70	ltled 11438	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
72	56	ltpnfd 13184	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < +∞)
73	52, 54, 55, 71, 72	elicod 13457	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ (𝐾[,)+∞))
74	51, 47, 73	fnfvimad 7271	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) ∈ (𝐹 “ (𝐾[,)+∞)))
75	49, 74	eqeltrrd 2845	. . 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 16400	. . . . . . 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 4559	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → if(2 ∥ 𝑛, 0, 1) = 1)
81	47	peano2nnd 12310	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℕ)
82		1xr 11349	. . . . . 6 ⊢ 1 ∈ ℝ^*
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ^*)
84	12, 80, 81, 83	fvmptd2 7037	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) = 1)
85	81	nnxrd 45188	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ^*)
86	81	nnred 12308	. . . . . . 7 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ)
87	56	ltp1d 12225	. . . . . . . 8 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
88	15, 56, 86, 70, 87	lttrd 11451	. . . . . . 7 ⊢ (𝜑 → 𝐾 < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
89	15, 86, 88	ltled 11438	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
90	86	ltpnfd 13184	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) < +∞)
91	52, 54, 85, 89, 90	elicod 13457	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ (𝐾[,)+∞))
92	51, 81, 91	fnfvimad 7271	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) ∈ (𝐹 “ (𝐾[,)+∞)))
93	84, 92	eqeltrrd 2845	. . 3 ⊢ (𝜑 → 1 ∈ (𝐹 “ (𝐾[,)+∞)))
94	75, 93	prssd 4847	. 2 ⊢ (𝜑 → {0, 1} ⊆ (𝐹 “ (𝐾[,)+∞)))
95	14, 94	eqssd 4026	1 ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ifcif 4548 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 “ cima 5703 Fn wfn 6568 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 ℕcn 12293 2c2 12348 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 [,)cico 13409 ⌈cceil 13842 ∥ cdvds 16302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ico 13413 df-fl 13843 df-ceil 13844 df-dvds 16303

This theorem is referenced by: limsup10ex 45694 liminf10ex 45695

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