limsup10exlem - Mathbox for Glauco Siliprandi

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Theorem limsup10exlem 44003

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 11149	. . . . . . 7 ⊢ 0 ∈ V
2	1	prid1 4723	. . . . . 6 ⊢ 0 ∈ {0, 1}
3		1re 11155	. . . . . . . 8 ⊢ 1 ∈ ℝ
4	3	elexi 3464	. . . . . . 7 ⊢ 1 ∈ V
5	4	prid2 4724	. . . . . 6 ⊢ 1 ∈ {0, 1}
6	2, 5	ifcli 4533	. . . . 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 3143	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))if(2 ∥ 𝑛, 0, 1) ∈ {0, 1})
9		nfv 1917	. . . 4 ⊢ Ⅎ𝑛𝜑
10	1, 4	ifex 4536	. . . . 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 43481	. . 3 ⊢ (𝜑 → ((𝐹 “ (𝐾[,)+∞)) ⊆ {0, 1} ↔ ∀𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))if(2 ∥ 𝑛, 0, 1) ∈ {0, 1}))
14	8, 13	mpbird 256	. 2 ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) ⊆ {0, 1})
15		limsup10exlem.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
16	15	ceilcld 13748	. . . . . . . . 9 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
17		1zzd 12534	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
18	16, 17	ifcld 4532	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ)
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ)
20		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
21		2teven 16237	. . . . . . 7 ⊢ ((if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → 2 ∥ 𝑛)
22	19, 20, 21	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → 2 ∥ 𝑛)
23	22	iftrued 4494	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → if(2 ∥ 𝑛, 0, 1) = 0)
24		2nn 12226	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
26		eqid 2736	. . . . . . . 8 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
27	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ∈ ℝ)
28	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ∈ ℝ)
29	16	zred 12607	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
30	29	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → (⌈‘𝐾) ∈ ℝ)
31		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ 𝐾)
32	15	ceilged 13751	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
33	32	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ≤ (⌈‘𝐾))
34	27, 28, 30, 31, 33	letrd 11312	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ (⌈‘𝐾))
35		iftrue 4492	. . . . . . . . . . 11 ⊢ (1 ≤ 𝐾 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = (⌈‘𝐾))
36	35	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = (⌈‘𝐾))
37	34, 36	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
38	3	leidi 11689	. . . . . . . . . . 11 ⊢ 1 ≤ 1
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ 1)
40		iffalse 4495	. . . . . . . . . . 11 ⊢ (¬ 1 ≤ 𝐾 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = 1)
41	40	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) = 1)
42	39, 41	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
43	37, 42	pm2.61dan 811	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
44	26, 17, 18, 43	eluzd 43634	. . . . . . 7 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ (ℤ_≥‘1))
45		nnuz 12806	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
46	44, 45	eleqtrrdi 2849	. . . . . 6 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℕ)
47	25, 46	nnmulcld 12206	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℕ)
48	1	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ V)
49	12, 23, 47, 48	fvmptd2 6956	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) = 0)
50	10, 12	fnmpti 6644	. . . . . 6 ⊢ 𝐹 Fn ℕ
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 Fn ℕ)
52	15	rexrd 11205	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
53		pnfxr 11209	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
55	47	nnxrd 43497	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ^*)
56	47	nnred 12168	. . . . . . 7 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ)
57	46	nnred 12168	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ)
58	33, 36	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
59	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ∈ ℝ)
60	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ∈ ℝ)
61		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → ¬ 1 ≤ 𝐾)
62	59, 60, 61	nleltd 43677	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 < 1)
63	59, 60, 62	ltled 11303	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ 1)
64	41	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 = if(1 ≤ 𝐾, (⌈‘𝐾), 1))
65	63, 64	breqtrd 5131	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
66	58, 65	pm2.61dan 811	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
67	46	nnrpd 12955	. . . . . . . . 9 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ⁺)
68		2timesgt 43512	. . . . . . . . 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 11313	. . . . . . 7 ⊢ (𝜑 → 𝐾 < (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
71	15, 56, 70	ltled 11303	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
72	56	ltpnfd 13042	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < +∞)
73	52, 54, 55, 71, 72	elicod 13314	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ (𝐾[,)+∞))
74	51, 47, 73	fnfvimad 7184	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) ∈ (𝐹 “ (𝐾[,)+∞)))
75	49, 74	eqeltrrd 2839	. . 3 ⊢ (𝜑 → 0 ∈ (𝐹 “ (𝐾[,)+∞)))
76	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ)
77		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
78		2tp1odd 16234	. . . . . . 7 ⊢ ((if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℤ ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → ¬ 2 ∥ 𝑛)
79	76, 77, 78	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → ¬ 2 ∥ 𝑛)
80	79	iffalsed 4497	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → if(2 ∥ 𝑛, 0, 1) = 1)
81	47	peano2nnd 12170	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℕ)
82		1xr 11214	. . . . . 6 ⊢ 1 ∈ ℝ^*
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ^*)
84	12, 80, 81, 83	fvmptd2 6956	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) = 1)
85	81	nnxrd 43497	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ^*)
86	81	nnred 12168	. . . . . . 7 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ)
87	56	ltp1d 12085	. . . . . . . 8 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
88	15, 56, 86, 70, 87	lttrd 11316	. . . . . . 7 ⊢ (𝜑 → 𝐾 < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
89	15, 86, 88	ltled 11303	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
90	86	ltpnfd 13042	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) < +∞)
91	52, 54, 85, 89, 90	elicod 13314	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ (𝐾[,)+∞))
92	51, 81, 91	fnfvimad 7184	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) ∈ (𝐹 “ (𝐾[,)+∞)))
93	84, 92	eqeltrrd 2839	. . 3 ⊢ (𝜑 → 1 ∈ (𝐹 “ (𝐾[,)+∞)))
94	75, 93	prssd 4782	. 2 ⊢ (𝜑 → {0, 1} ⊆ (𝐹 “ (𝐾[,)+∞)))
95	14, 94	eqssd 3961	1 ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ifcif 4486 {cpr 4588 class class class wbr 5105 ↦ cmpt 5188 “ cima 5636 Fn wfn 6491 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 ℕcn 12153 2c2 12208 ℤcz 12499 ℤ_≥cuz 12763 ℝ⁺crp 12915 [,)cico 13266 ⌈cceil 13696 ∥ cdvds 16136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-ico 13270 df-fl 13697 df-ceil 13698 df-dvds 16137

This theorem is referenced by: limsup10ex 44004 liminf10ex 44005

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