limsup10exlem - Mathbox for Glauco Siliprandi

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Theorem limsup10exlem 43649

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 11070	. . . . . . 7 ⊢ 0 ∈ V
2	1	prid1 4710	. . . . . 6 ⊢ 0 ∈ {0, 1}
3		1re 11076	. . . . . . . 8 ⊢ 1 ∈ ℝ
4	3	elexi 3460	. . . . . . 7 ⊢ 1 ∈ V
5	4	prid2 4711	. . . . . 6 ⊢ 1 ∈ {0, 1}
6	2, 5	ifcli 4520	. . . . 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 3139	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∩ (𝐾[,)+∞))if(2 ∥ 𝑛, 0, 1) ∈ {0, 1})
9		nfv 1916	. . . 4 ⊢ Ⅎ𝑛𝜑
10	1, 4	ifex 4523	. . . . 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 43138	. . 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 13664	. . . . . . . . 9 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
17		1zzd 12452	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
18	16, 17	ifcld 4519	. . . . . . . 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 16163	. . . . . . 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 4481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) → if(2 ∥ 𝑛, 0, 1) = 0)
24		2nn 12147	. . . . . . 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 12527	. . . . . . . . . . . 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 13667	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
33	32	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 𝐾 ≤ (⌈‘𝐾))
34	27, 28, 30, 31, 33	letrd 11233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ (⌈‘𝐾))
35		iftrue 4479	. . . . . . . . . . 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 5120	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
38	3	leidi 11610	. . . . . . . . . . 11 ⊢ 1 ≤ 1
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ 1)
40		iffalse 4482	. . . . . . . . . . 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 5120	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
43	37, 42	pm2.61dan 810	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
44	26, 17, 18, 43	eluzd 43284	. . . . . . 7 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ (ℤ_≥‘1))
45		nnuz 12722	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
46	44, 45	eleqtrrdi 2848	. . . . . 6 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℕ)
47	25, 46	nnmulcld 12127	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℕ)
48	1	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ V)
49	12, 23, 47, 48	fvmptd2 6939	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) = 0)
50	10, 12	fnmpti 6627	. . . . . 6 ⊢ 𝐹 Fn ℕ
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 Fn ℕ)
52	15	rexrd 11126	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
53		pnfxr 11130	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
55	47	nnxrd 42905	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ^*)
56	47	nnred 12089	. . . . . . 7 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ ℝ)
57	46	nnred 12089	. . . . . . . 8 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ)
58	33, 36	breqtrrd 5120	. . . . . . . . 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 43327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 < 1)
63	59, 60, 62	ltled 11224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ 1)
64	41	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 1 = if(1 ≤ 𝐾, (⌈‘𝐾), 1))
65	63, 64	breqtrd 5118	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 ≤ 𝐾) → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
66	58, 65	pm2.61dan 810	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ≤ if(1 ≤ 𝐾, (⌈‘𝐾), 1))
67	46	nnrpd 12871	. . . . . . . . 9 ⊢ (𝜑 → if(1 ≤ 𝐾, (⌈‘𝐾), 1) ∈ ℝ⁺)
68		2timesgt 43162	. . . . . . . . 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 11234	. . . . . . 7 ⊢ (𝜑 → 𝐾 < (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
71	15, 56, 70	ltled 11224	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)))
72	56	ltpnfd 12958	. . . . . 6 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < +∞)
73	52, 54, 55, 71, 72	elicod 13230	. . . . 5 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) ∈ (𝐾[,)+∞))
74	51, 47, 73	fnfvimad 7166	. . . 4 ⊢ (𝜑 → (𝐹‘(2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1))) ∈ (𝐹 “ (𝐾[,)+∞)))
75	49, 74	eqeltrrd 2838	. . 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 16160	. . . . . . 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 4484	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) → if(2 ∥ 𝑛, 0, 1) = 1)
81	47	peano2nnd 12091	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℕ)
82		1xr 11135	. . . . . 6 ⊢ 1 ∈ ℝ^*
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ^*)
84	12, 80, 81, 83	fvmptd2 6939	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) = 1)
85	81	nnxrd 42905	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ^*)
86	81	nnred 12089	. . . . . . 7 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ ℝ)
87	56	ltp1d 12006	. . . . . . . 8 ⊢ (𝜑 → (2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
88	15, 56, 86, 70, 87	lttrd 11237	. . . . . . 7 ⊢ (𝜑 → 𝐾 < ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
89	15, 86, 88	ltled 11224	. . . . . 6 ⊢ (𝜑 → 𝐾 ≤ ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1))
90	86	ltpnfd 12958	. . . . . 6 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) < +∞)
91	52, 54, 85, 89, 90	elicod 13230	. . . . 5 ⊢ (𝜑 → ((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1) ∈ (𝐾[,)+∞))
92	51, 81, 91	fnfvimad 7166	. . . 4 ⊢ (𝜑 → (𝐹‘((2 · if(1 ≤ 𝐾, (⌈‘𝐾), 1)) + 1)) ∈ (𝐹 “ (𝐾[,)+∞)))
93	84, 92	eqeltrrd 2838	. . 3 ⊢ (𝜑 → 1 ∈ (𝐹 “ (𝐾[,)+∞)))
94	75, 93	prssd 4769	. 2 ⊢ (𝜑 → {0, 1} ⊆ (𝐹 “ (𝐾[,)+∞)))
95	14, 94	eqssd 3949	1 ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 Vcvv 3441 ∩ cin 3897 ⊆ wss 3898 ifcif 4473 {cpr 4575 class class class wbr 5092 ↦ cmpt 5175 “ cima 5623 Fn wfn 6474 ‘cfv 6479 (class class class)co 7337 ℝcr 10971 0cc0 10972 1c1 10973 + caddc 10975 · cmul 10977 +∞cpnf 11107 ℝ^*cxr 11109 < clt 11110 ≤ cle 11111 ℕcn 12074 2c2 12129 ℤcz 12420 ℤ_≥cuz 12683 ℝ⁺crp 12831 [,)cico 13182 ⌈cceil 13612 ∥ cdvds 16062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-n0 12335 df-z 12421 df-uz 12684 df-rp 12832 df-ico 13186 df-fl 13613 df-ceil 13614 df-dvds 16063

This theorem is referenced by: limsup10ex 43650 liminf10ex 43651

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