limsupval2 - Metamath Proof Explorer

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Theorem limsupval2 15387

Description: The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
limsupval.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
limsupval2.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
limsupval2.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupval2.3	⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)

**Assertion**
Ref	Expression
limsupval2	⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ^*, < ))

Distinct variable groups: 𝑘,𝐹 𝐴,𝑘 Allowed substitution hints: 𝜑(𝑘) 𝐺(𝑘) 𝑉(𝑘)

Proof of Theorem limsupval2 Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupval2.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		limsupval.1	. . . 4 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
3	2	limsupval 15381	. . 3 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ^*, < ))
4	1, 3	syl 17	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ^*, < ))
5		imassrn 6020	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
6	2	limsupgf 15382	. . . . . . 7 ⊢ 𝐺:ℝ⟶ℝ^*
7		frn 6658	. . . . . . 7 ⊢ (𝐺:ℝ⟶ℝ^* → ran 𝐺 ⊆ ℝ^*)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran 𝐺 ⊆ ℝ^*
9		infxrlb 13234	. . . . . . 7 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
10	9	ralrimiva 3124	. . . . . 6 ⊢ (ran 𝐺 ⊆ ℝ^* → ∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
11	8, 10	mp1i 13	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
12		ssralv 4003	. . . . 5 ⊢ ((𝐺 “ 𝐴) ⊆ ran 𝐺 → (∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ^, < ) ≤ 𝑥 → ∀𝑥 ∈ (𝐺 “ 𝐴)inf(ran 𝐺, ℝ^, < ) ≤ 𝑥))
13	5, 11, 12	mpsyl 68	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐺 “ 𝐴)inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
14	5, 8	sstri 3944	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ℝ^*
15		infxrcl 13233	. . . . . 6 ⊢ (ran 𝐺 ⊆ ℝ^* → inf(ran 𝐺, ℝ^, < ) ∈ ℝ^)
16	8, 15	ax-mp 5	. . . . 5 ⊢ inf(ran 𝐺, ℝ^, < ) ∈ ℝ^
17		infxrgelb 13235	. . . . 5 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ^* ∧ inf(ran 𝐺, ℝ^, < ) ∈ ℝ^) → (inf(ran 𝐺, ℝ^, < ) ≤ inf((𝐺 “ 𝐴), ℝ^, < ) ↔ ∀𝑥 ∈ (𝐺 “ 𝐴)inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥))
18	14, 16, 17	mp2an 692	. . . 4 ⊢ (inf(ran 𝐺, ℝ^, < ) ≤ inf((𝐺 “ 𝐴), ℝ^, < ) ↔ ∀𝑥 ∈ (𝐺 “ 𝐴)inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
19	13, 18	sylibr 234	. . 3 ⊢ (𝜑 → inf(ran 𝐺, ℝ^, < ) ≤ inf((𝐺 “ 𝐴), ℝ^, < ))
20		limsupval2.3	. . . . . . 7 ⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
21		limsupval2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
22		ressxr 11156	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
23	21, 22	sstrdi 3947	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
24		supxrunb1 13218	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
26	20, 25	mpbird 257	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥)
27		infxrcl 13233	. . . . . . . . . 10 ⊢ ((𝐺 “ 𝐴) ⊆ ℝ^* → inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^)
28	14, 27	mp1i 13	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^)
29	21	sselda 3934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	29	ad2ant2r 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
31	6	ffvelcdmi 7016	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) ∈ ℝ^*)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ ℝ^*)
33	6	ffvelcdmi 7016	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ^*)
34	33	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ^*)
35		ffn 6651	. . . . . . . . . . . 12 ⊢ (𝐺:ℝ⟶ℝ^* → 𝐺 Fn ℝ)
36	6, 35	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝐺 Fn ℝ)
37	21	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝐴 ⊆ ℝ)
38		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
39		fnfvima 7167	. . . . . . . . . . 11 ⊢ ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
40	36, 37, 38, 39	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
41		infxrlb 13234	. . . . . . . . . 10 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ^* ∧ (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
42	14, 40, 41	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
43		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ)
44		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
45		limsupgord 15379	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
46	43, 30, 44, 45	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
47	2	limsupgval 15383	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ))
48	30, 47	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ))
49	2	limsupgval 15383	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
50	49	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
51	46, 48, 50	3brtr4d 5123	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ (𝐺‘𝑛))
52	28, 32, 34, 42, 51	xrletrd 13061	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
53	52	rexlimdvaa 3134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
54	53	ralimdva 3144	. . . . . 6 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → ∀𝑛 ∈ ℝ inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
55	26, 54	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℝ inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
56	6, 35	ax-mp 5	. . . . . 6 ⊢ 𝐺 Fn ℝ
57		breq2 5095	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛)))
58	57	ralrn 7021	. . . . . 6 ⊢ (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ℝ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛)))
59	56, 58	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ℝ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛))
60	55, 59	sylibr 234	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ 𝑥)
61	14, 27	ax-mp 5	. . . . 5 ⊢ inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^
62		infxrgelb 13235	. . . . 5 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^) → (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ 𝑥))
63	8, 61, 62	mp2an 692	. . . 4 ⊢ (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ 𝑥)
64	60, 63	sylibr 234	. . 3 ⊢ (𝜑 → inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < ))
65		xrletri3 13053	. . . 4 ⊢ ((inf(ran 𝐺, ℝ^, < ) ∈ ℝ^ ∧ inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^) → (inf(ran 𝐺, ℝ^, < ) = inf((𝐺 “ 𝐴), ℝ^, < ) ↔ (inf(ran 𝐺, ℝ^, < ) ≤ inf((𝐺 “ 𝐴), ℝ^, < ) ∧ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < ))))
66	16, 61, 65	mp2an 692	. . 3 ⊢ (inf(ran 𝐺, ℝ^, < ) = inf((𝐺 “ 𝐴), ℝ^, < ) ↔ (inf(ran 𝐺, ℝ^, < ) ≤ inf((𝐺 “ 𝐴), ℝ^, < ) ∧ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < )))
67	19, 64, 66	sylanbrc 583	. 2 ⊢ (𝜑 → inf(ran 𝐺, ℝ^, < ) = inf((𝐺 “ 𝐴), ℝ^, < ))
68	4, 67	eqtrd 2766	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ∩ cin 3901 ⊆ wss 3902 class class class wbr 5091 ↦ cmpt 5172 ran crn 5617 “ cima 5619 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 supcsup 9324 infcinf 9325 ℝcr 11005 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 [,)cico 13247 lim supclsp 15377

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-ico 13251 df-limsup 15378

This theorem is referenced by: mbflimsup 25595 limsupresico 45744 limsupvaluz 45752

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