limsupval2 - Metamath Proof Explorer

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Theorem limsupval2 15519

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 15513	. . 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 6063	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
6	2	limsupgf 15514	. . . . . . 7 ⊢ 𝐺:ℝ⟶ℝ^*
7		frn 6703	. . . . . . 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 3157	. . . . . 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 4008	. . . . 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 3948	. . . . 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 704	. . . 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 11241	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
23	21, 22	sstrdi 3951	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
24		supxrunb1 13333	. . . . . . . 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 13348	. . . . . . . . . 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 3939	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	29	ad2ant2r 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
31	6	ffvelcdmi 7068	. . . . . . . . . 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 7068	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ^*)
34	33	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ^*)
35		ffn 6695	. . . . . . . . . . . 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 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝐴 ⊆ ℝ)
38		simprl 782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
39		fnfvima 7221	. . . . . . . . . . 11 ⊢ ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
40	36, 37, 38, 39	syl3anc 1394	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
43		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ)
44		simprr 784	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
45		limsupgord 15511	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
46	43, 30, 44, 45	syl3anc 1394	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
47	2	limsupgval 15515	. . . . . . . . . . 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 15515	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
50	49	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
51	46, 48, 50	3brtr4d 5136	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ (𝐺‘𝑛))
52	28, 32, 34, 42, 51	xrletrd 13175	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
53	52	rexlimdvaa 3167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
54	53	ralimdva 3177	. . . . . 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 5108	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛)))
58	57	ralrn 7073	. . . . . 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 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 704	. . . 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 13167	. . . 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 704	. . 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 594	. 2 ⊢ (𝜑 → inf(ran 𝐺, ℝ^, < ) = inf((𝐺 “ 𝐴), ℝ^, < ))
68	4, 67	eqtrd 2800	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ∃wrex 3089 ∩ cin 3906 ⊆ wss 3907 class class class wbr 5104 ↦ cmpt 5185 ran crn 5652 “ cima 5654 Fn wfn 6520 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 supcsup 9388 infcinf 9389 ℝcr 11087 +∞cpnf 11228 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232 [,)cico 13362 lim supclsp 15509

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7974 df-2nd 7975 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-ico 13366 df-limsup 15510

This theorem is referenced by: mbflimsup 25782 limsupresico 46273 limsupvaluz 46281

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