limsupval2 - Metamath Proof Explorer

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Theorem limsupval2 15567

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 15561	. . 3 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ^*, < ))
4	1, 3	syl 18	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ^*, < ))
5		imassrn 6067	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
6	2	limsupgf 15562	. . . . . . 7 ⊢ 𝐺:ℝ⟶ℝ^*
7		frn 6710	. . . . . . 7 ⊢ (𝐺:ℝ⟶ℝ^* → ran 𝐺 ⊆ ℝ^*)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran 𝐺 ⊆ ℝ^*
9		infxrlb 13387	. . . . . . 7 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
10	9	ralrimiva 3154	. . . . . 6 ⊢ (ran 𝐺 ⊆ ℝ^* → ∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
11	8, 10	mp1i 14	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
12		ssralv 4000	. . . . 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 69	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐺 “ 𝐴)inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
14	5, 8	sstri 3940	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ℝ^*
15		infxrcl 13386	. . . . . 6 ⊢ (ran 𝐺 ⊆ ℝ^* → inf(ran 𝐺, ℝ^, < ) ∈ ℝ^)
16	8, 15	ax-mp 5	. . . . 5 ⊢ inf(ran 𝐺, ℝ^, < ) ∈ ℝ^
17		infxrgelb 13388	. . . . 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 705	. . . 4 ⊢ (inf(ran 𝐺, ℝ^, < ) ≤ inf((𝐺 “ 𝐴), ℝ^, < ) ↔ ∀𝑥 ∈ (𝐺 “ 𝐴)inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
19	13, 18	sylibr 237	. . 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 11277	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
23	21, 22	sstrdi 3943	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
24		supxrunb1 13371	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
25	23, 24	syl 18	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
26	20, 25	mpbird 260	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥)
27		infxrcl 13386	. . . . . . . . . 10 ⊢ ((𝐺 “ 𝐴) ⊆ ℝ^* → inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^)
28	14, 27	mp1i 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^)
29	21	sselda 3931	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	29	ad2ant2r 760	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
31	6	ffvelcdmi 7076	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) ∈ ℝ^*)
32	30, 31	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ ℝ^*)
33	6	ffvelcdmi 7076	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ^*)
34	33	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ^*)
35		ffn 6702	. . . . . . . . . . . 12 ⊢ (𝐺:ℝ⟶ℝ^* → 𝐺 Fn ℝ)
36	6, 35	mp1i 14	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝐺 Fn ℝ)
37	21	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝐴 ⊆ ℝ)
38		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
39		fnfvima 7232	. . . . . . . . . . 11 ⊢ ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
40	36, 37, 38, 39	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
41		infxrlb 13387	. . . . . . . . . 10 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ^* ∧ (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
42	14, 40, 41	sylancr 599	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
43		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ)
44		simprr 785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
45		limsupgord 15559	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
46	43, 30, 44, 45	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
47	2	limsupgval 15563	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ))
48	30, 47	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) = sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ))
49	2	limsupgval 15563	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
50	49	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
51	46, 48, 50	3brtr4d 5137	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ (𝐺‘𝑛))
52	28, 32, 34, 42, 51	xrletrd 13213	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
53	52	rexlimdvaa 3164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
54	53	ralimdva 3174	. . . . . 6 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → ∀𝑛 ∈ ℝ inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
55	26, 54	mpd 16	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℝ inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
56	6, 35	ax-mp 5	. . . . . 6 ⊢ 𝐺 Fn ℝ
57		breq2 5107	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛)))
58	57	ralrn 7081	. . . . . 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 237	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ 𝑥)
61	14, 27	ax-mp 5	. . . . 5 ⊢ inf((𝐺 “ 𝐴), ℝ^, < ) ∈ ℝ^
62		infxrgelb 13388	. . . . 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 705	. . . 4 ⊢ (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ 𝑥)
64	60, 63	sylibr 237	. . 3 ⊢ (𝜑 → inf((𝐺 “ 𝐴), ℝ^, < ) ≤ inf(ran 𝐺, ℝ^, < ))
65		xrletri3 13205	. . . 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 705	. . 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 595	. 2 ⊢ (𝜑 → inf(ran 𝐺, ℝ^, < ) = inf((𝐺 “ 𝐴), ℝ^, < ))
68	4, 67	eqtrd 2795	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ^*, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 “ cima 5658 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 supcsup 9410 infcinf 9411 ℝcr 11123 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 [,)cico 13400 lim supclsp 15557

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7986 df-2nd 7987 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-ico 13404 df-limsup 15558

This theorem is used by: mbflimsup 25894 limsupresico 46528 limsupvaluz 46536

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