Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupresuz | Structured version Visualization version GIF version |
Description: If the real part of the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupresuz.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupresuz.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
limsupresuz.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
limsupresuz.d | ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) |
Ref | Expression |
---|---|
limsupresuz | ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescom 5872 | . . . . 5 ⊢ ((𝐹 ↾ 𝑍) ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ 𝑍) | |
2 | 1 | fveq2i 6666 | . . . 4 ⊢ (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍))) |
4 | relres 5875 | . . . . . . . . . 10 ⊢ Rel (𝐹 ↾ ℝ) | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel (𝐹 ↾ ℝ)) |
6 | limsupresuz.d | . . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) | |
7 | relssres 5886 | . . . . . . . . 9 ⊢ ((Rel (𝐹 ↾ ℝ) ∧ dom (𝐹 ↾ ℝ) ⊆ ℤ) → ((𝐹 ↾ ℝ) ↾ ℤ) = (𝐹 ↾ ℝ)) | |
8 | 5, 6, 7 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ ℤ) = (𝐹 ↾ ℝ)) |
9 | 8 | eqcomd 2824 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ ℤ)) |
10 | 9 | reseq1d 5845 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)) = (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞))) |
11 | resres 5859 | . . . . . . 7 ⊢ (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞)) = ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞))) | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞)) = ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞)))) |
13 | limsupresuz.m | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
14 | limsupresuz.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
15 | 13, 14 | uzinico 41712 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) |
16 | 15 | eqcomd 2824 | . . . . . . 7 ⊢ (𝜑 → (ℤ ∩ (𝑀[,)+∞)) = 𝑍) |
17 | 16 | reseq2d 5846 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞))) = ((𝐹 ↾ ℝ) ↾ 𝑍)) |
18 | 10, 12, 17 | 3eqtrrd 2858 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ 𝑍) = ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) |
19 | 18 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim sup‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)))) |
20 | 13 | zred 12075 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
21 | eqid 2818 | . . . . 5 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
22 | limsupresuz.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
23 | 22 | resexd 41279 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ V) |
24 | 20, 21, 23 | limsupresico 41857 | . . . 4 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) = (lim sup‘(𝐹 ↾ ℝ))) |
25 | 19, 24 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim sup‘(𝐹 ↾ ℝ))) |
26 | 3, 25 | eqtrd 2853 | . 2 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘(𝐹 ↾ ℝ))) |
27 | 22 | resexd 41279 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑍) ∈ V) |
28 | 27 | limsupresre 41853 | . 2 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘(𝐹 ↾ 𝑍))) |
29 | 22 | limsupresre 41853 | . 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹)) |
30 | 26, 28, 29 | 3eqtr3d 2861 | 1 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 +∞cpnf 10660 ℤcz 11969 ℤ≥cuz 12231 [,)cico 12728 lim supclsp 14815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ico 12732 df-limsup 14816 |
This theorem is referenced by: limsupresuz2 41866 |
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