limsupval2 - Metamath Proof Explorer

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Theorem limsupval2 15433

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 15427	. . 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 6030	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
6	2	limsupgf 15428	. . . . . . 7 ⊢ 𝐺:ℝ⟶ℝ^*
7		frn 6669	. . . . . . 7 ⊢ (𝐺:ℝ⟶ℝ^* → ran 𝐺 ⊆ ℝ^*)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ran 𝐺 ⊆ ℝ^*
9		infxrlb 13278	. . . . . . 7 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝐺) → inf(ran 𝐺, ℝ^*, < ) ≤ 𝑥)
10	9	ralrimiva 3130	. . . . . 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 3991	. . . . 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 3932	. . . . 5 ⊢ (𝐺 “ 𝐴) ⊆ ℝ^*
15		infxrcl 13277	. . . . . 6 ⊢ (ran 𝐺 ⊆ ℝ^* → inf(ran 𝐺, ℝ^, < ) ∈ ℝ^)
16	8, 15	ax-mp 5	. . . . 5 ⊢ inf(ran 𝐺, ℝ^, < ) ∈ ℝ^
17		infxrgelb 13279	. . . . 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 693	. . . 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 11180	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
23	21, 22	sstrdi 3935	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
24		supxrunb1 13262	. . . . . . . 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 13277	. . . . . . . . . 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 3922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
30	29	ad2ant2r 748	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
31	6	ffvelcdmi 7029	. . . . . . . . . 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 7029	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ^*)
34	33	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ^*)
35		ffn 6662	. . . . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝐴 ⊆ ℝ)
38		simprl 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
39		fnfvima 7181	. . . . . . . . . . 11 ⊢ ((𝐺 Fn ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
40	36, 37, 38, 39	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
41		infxrlb 13278	. . . . . . . . . 10 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ^* ∧ (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
42	14, 40, 41	sylancr 588	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑥))
43		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ)
44		simprr 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
45		limsupgord 15425	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
46	43, 30, 44, 45	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → sup(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
47	2	limsupgval 15429	. . . . . . . . . . 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 15429	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
50	49	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
51	46, 48, 50	3brtr4d 5118	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ (𝐺‘𝑛))
52	28, 32, 34, 42, 51	xrletrd 13104	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛))
53	52	rexlimdvaa 3140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → inf((𝐺 “ 𝐴), ℝ^*, < ) ≤ (𝐺‘𝑛)))
54	53	ralimdva 3150	. . . . . 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 5090	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (inf((𝐺 “ 𝐴), ℝ^, < ) ≤ 𝑥 ↔ inf((𝐺 “ 𝐴), ℝ^, < ) ≤ (𝐺‘𝑛)))
58	57	ralrn 7034	. . . . . 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 13279	. . . . 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 693	. . . 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 13096	. . . 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 693	. . 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 584	. 2 ⊢ (𝜑 → inf(ran 𝐺, ℝ^, < ) = inf((𝐺 “ 𝐴), ℝ^, < ))
68	4, 67	eqtrd 2772	1 ⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ran crn 5625 “ cima 5627 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 supcsup 9346 infcinf 9347 ℝcr 11028 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 [,)cico 13291 lim supclsp 15423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-ico 13295 df-limsup 15424

This theorem is referenced by: mbflimsup 25643 limsupresico 46146 limsupvaluz 46154

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