Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfresre | Structured version Visualization version GIF version |
Description: The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfresre.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
liminfresre | ⊢ (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 12838 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 1 | resabs1i 41406 | . . . 4 ⊢ ((𝐹 ↾ ℝ) ↾ (0[,)+∞)) = (𝐹 ↾ (0[,)+∞)) |
3 | 2 | fveq2i 6668 | . . 3 ⊢ (lim inf‘((𝐹 ↾ ℝ) ↾ (0[,)+∞))) = (lim inf‘(𝐹 ↾ (0[,)+∞))) |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ ℝ) ↾ (0[,)+∞))) = (lim inf‘(𝐹 ↾ (0[,)+∞)))) |
5 | 0red 10638 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
6 | eqid 2821 | . . 3 ⊢ (0[,)+∞) = (0[,)+∞) | |
7 | liminfresre.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
8 | 7 | resexd 41395 | . . 3 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ V) |
9 | 5, 6, 8 | liminfresico 42044 | . 2 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ ℝ) ↾ (0[,)+∞))) = (lim inf‘(𝐹 ↾ ℝ))) |
10 | 5, 6, 7 | liminfresico 42044 | . 2 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ (0[,)+∞))) = (lim inf‘𝐹)) |
11 | 4, 9, 10 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 +∞cpnf 10666 [,)cico 12734 lim infclsi 42024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-ico 12738 df-liminf 42025 |
This theorem is referenced by: liminfresuz 42057 |
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