limsupval2 - Metamath Proof Explorer

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Theorem limsupval2 15494

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 15488	. . 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 6058	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
6	2	limsupgf 15489	. . . . . . 7 ⊢ 𝐺:ℝ⟶ℝ^*
7		frn 6712	. . . . . . 7 ⊢ (𝐺:ℝ⟶ℝ^* → ran 𝐺 ⊆ ℝ^*)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran 𝐺 ⊆ ℝ^*
9		infxrlb 13349	. . . . . . 7 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
10	9	ralrimiva 3132	. . . . . 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 4027	. . . . 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 3968	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ℝ^*
15		infxrcl 13348	. . . . . 6 ⊢ (ran 𝐺 ⊆ ℝ^* → inf(ran 𝐺, ℝ^, < ) ∈ ℝ^)
16	8, 15	ax-mp 5	. . . . 5 ⊢ inf(ran 𝐺, ℝ^, < ) ∈ ℝ^
17		infxrgelb 13350	. . . . 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 11277	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
23	21, 22	sstrdi 3971	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
24		supxrunb1 13333	. . . . . . . 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 13348	. . . . . . . . . 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 3958	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	29	ad2ant2r 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
31	6	ffvelcdmi 7072	. . . . . . . . . 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 7072	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ^*)
34	33	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ^*)
35		ffn 6705	. . . . . . . . . . . 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 7224	. . . . . . . . . . 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 13349	. . . . . . . . . 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 15486	. . . . . . . . . . 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 15490	. . . . . . . . . . 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 15490	. . . . . . . . . . 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 5151	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ (𝐺‘𝑛))
52	28, 32, 34, 42, 51	xrletrd 13176	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
53	52	rexlimdvaa 3142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
54	53	ralimdva 3152	. . . . . 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 5123	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛)))
58	57	ralrn 7077	. . . . . 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 13350	. . . . 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 13168	. . . 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 2770	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 ∩ cin 3925 ⊆ wss 3926 class class class wbr 5119 ↦ cmpt 5201 ran crn 5655 “ cima 5657 Fn wfn 6525 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 supcsup 9450 infcinf 9451 ℝcr 11126 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 [,)cico 13362 lim supclsp 15484

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7986 df-2nd 7987 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9452 df-inf 9453 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-ico 13366 df-limsup 15485

This theorem is referenced by: mbflimsup 25617 limsupresico 45677 limsupvaluz 45685

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