liminfval - Mathbox for Glauco Siliprandi

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Theorem liminfval 46205

Description: The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypothesis**
Ref	Expression
liminfval.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))

**Assertion**
Ref	Expression
liminfval	⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ^*, < ))

Distinct variable group: 𝑘,𝐹 Allowed substitution hints: 𝐺(𝑘) 𝑉(𝑘)

Proof of Theorem liminfval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-liminf 46198	. 2 ⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
2		imaeq1 6014	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4160	. . . . . . 7 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9384	. . . . . 6 ⊢ (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
5	4	mpteq2dv 5180	. . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
6		liminfval.1	. . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
7	6	a1i 11	. . . . 5 ⊢ (𝑥 = 𝐹 → 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
8	5, 7	eqtr4d 2775	. . . 4 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = 𝐺)
9	8	rneqd 5887	. . 3 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran 𝐺)
10	9	supeq1d 9352	. 2 ⊢ (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = sup(ran 𝐺, ℝ^, < ))
11		elex 3451	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		xrltso 13083	. . . 4 ⊢ < Or ℝ^*
13	12	supex 9370	. . 3 ⊢ sup(ran 𝐺, ℝ^*, < ) ∈ V
14	13	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran 𝐺, ℝ^*, < ) ∈ V)
15	1, 10, 11, 14	fvmptd3 6965	1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ∩ cin 3889 ↦ cmpt 5167 ran crn 5625 “ cima 5627 ‘cfv 6492 (class class class)co 7360 supcsup 9346 infcinf 9347 ℝcr 11028 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 [,)cico 13291 lim infclsi 46197

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-pre-lttri 11103 ax-pre-lttrn 11104

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-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-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-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-liminf 46198

This theorem is referenced by: liminfcl 46209 liminfvald 46210 liminfval5 46211 liminfresxr 46213 liminfval2 46214 liminfvalxr 46229

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