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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 5967 | . . . . 5 ⊢ ((𝐹 ↾ 𝑍) ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ 𝑍) | |
2 | 1 | fveq2i 6849 | . . . 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 5970 | . . . . . . . . . 10 ⊢ Rel (𝐹 ↾ ℝ) | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel (𝐹 ↾ ℝ)) |
6 | limsupresuz.d | . . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) | |
7 | relssres 5982 | . . . . . . . . 9 ⊢ ((Rel (𝐹 ↾ ℝ) ∧ dom (𝐹 ↾ ℝ) ⊆ ℤ) → ((𝐹 ↾ ℝ) ↾ ℤ) = (𝐹 ↾ ℝ)) | |
8 | 5, 6, 7 | syl2anc 585 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ ℤ) = (𝐹 ↾ ℝ)) |
9 | 8 | eqcomd 2739 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ ℤ)) |
10 | 9 | reseq1d 5940 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)) = (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞))) |
11 | resres 5954 | . . . . . . 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 43888 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) |
16 | 15 | eqcomd 2739 | . . . . . . 7 ⊢ (𝜑 → (ℤ ∩ (𝑀[,)+∞)) = 𝑍) |
17 | 16 | reseq2d 5941 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞))) = ((𝐹 ↾ ℝ) ↾ 𝑍)) |
18 | 10, 12, 17 | 3eqtrrd 2778 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ 𝑍) = ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) |
19 | 18 | fveq2d 6850 | . . . 4 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim sup‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)))) |
20 | 13 | zred 12615 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
21 | eqid 2733 | . . . . 5 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
22 | limsupresuz.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
23 | 22 | resexd 5988 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ V) |
24 | 20, 21, 23 | limsupresico 44031 | . . . 4 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) = (lim sup‘(𝐹 ↾ ℝ))) |
25 | 19, 24 | eqtrd 2773 | . . 3 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim sup‘(𝐹 ↾ ℝ))) |
26 | 3, 25 | eqtrd 2773 | . 2 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘(𝐹 ↾ ℝ))) |
27 | 22 | resexd 5988 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑍) ∈ V) |
28 | 27 | limsupresre 44027 | . 2 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘(𝐹 ↾ 𝑍))) |
29 | 22 | limsupresre 44027 | . 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹)) |
30 | 26, 28, 29 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 dom cdm 5637 ↾ cres 5639 Rel wrel 5642 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 +∞cpnf 11194 ℤcz 12507 ℤ≥cuz 12771 [,)cico 13275 lim supclsp 15361 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-ico 13279 df-limsup 15362 |
This theorem is referenced by: limsupresuz2 44040 |
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