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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfresuz | Structured version Visualization version GIF version |
Description: If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfresuz.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminfresuz.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminfresuz.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
liminfresuz.d | ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) |
Ref | Expression |
---|---|
liminfresuz | ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescom 5842 | . . . . 5 ⊢ ((𝐹 ↾ 𝑍) ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ 𝑍) | |
2 | 1 | fveq2i 6654 | . . . 4 ⊢ (lim inf‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim inf‘((𝐹 ↾ ℝ) ↾ 𝑍)) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim inf‘((𝐹 ↾ ℝ) ↾ 𝑍))) |
4 | relres 5845 | . . . . . . . . . 10 ⊢ Rel (𝐹 ↾ ℝ) | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel (𝐹 ↾ ℝ)) |
6 | liminfresuz.d | . . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) | |
7 | relssres 5857 | . . . . . . . . 9 ⊢ ((Rel (𝐹 ↾ ℝ) ∧ dom (𝐹 ↾ ℝ) ⊆ ℤ) → ((𝐹 ↾ ℝ) ↾ ℤ) = (𝐹 ↾ ℝ)) | |
8 | 5, 6, 7 | syl2anc 588 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ ℤ) = (𝐹 ↾ ℝ)) |
9 | 8 | eqcomd 2765 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ ℤ)) |
10 | 9 | reseq1d 5815 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)) = (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞))) |
11 | resres 5829 | . . . . . . 7 ⊢ (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞)) = ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞))) | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞)) = ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞)))) |
13 | liminfresuz.m | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
14 | liminfresuz.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
15 | 13, 14 | uzinico 42548 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) |
16 | 15 | eqcomd 2765 | . . . . . . 7 ⊢ (𝜑 → (ℤ ∩ (𝑀[,)+∞)) = 𝑍) |
17 | 16 | reseq2d 5816 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞))) = ((𝐹 ↾ ℝ) ↾ 𝑍)) |
18 | 10, 12, 17 | 3eqtrrd 2799 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ 𝑍) = ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) |
19 | 18 | fveq2d 6655 | . . . 4 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim inf‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)))) |
20 | 13 | zred 12111 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
21 | eqid 2759 | . . . . 5 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
22 | liminfresuz.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
23 | 22 | resexd 5863 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ V) |
24 | 20, 21, 23 | liminfresico 42764 | . . . 4 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) = (lim inf‘(𝐹 ↾ ℝ))) |
25 | 19, 24 | eqtrd 2794 | . . 3 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim inf‘(𝐹 ↾ ℝ))) |
26 | 3, 25 | eqtrd 2794 | . 2 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim inf‘(𝐹 ↾ ℝ))) |
27 | 22 | resexd 5863 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑍) ∈ V) |
28 | 27 | liminfresre 42772 | . 2 ⊢ (𝜑 → (lim inf‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim inf‘(𝐹 ↾ 𝑍))) |
29 | 22 | liminfresre 42772 | . 2 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹)) |
30 | 26, 28, 29 | 3eqtr3d 2802 | 1 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∩ cin 3853 ⊆ wss 3854 dom cdm 5517 ↾ cres 5519 Rel wrel 5522 ‘cfv 6328 (class class class)co 7143 ℝcr 10559 +∞cpnf 10695 ℤcz 12005 ℤ≥cuz 12267 [,)cico 12766 lim infclsi 42744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-sup 8924 df-inf 8925 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-q 12374 df-ico 12770 df-liminf 42745 |
This theorem is referenced by: liminfresuz2 42780 |
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