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 5917 | . . . . 5 ⊢ ((𝐹 ↾ 𝑍) ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ 𝑍) | |
2 | 1 | fveq2i 6777 | . . . 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 5920 | . . . . . . . . . 10 ⊢ Rel (𝐹 ↾ ℝ) | |
5 | 4 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel (𝐹 ↾ ℝ)) |
6 | limsupresuz.d | . . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) | |
7 | relssres 5932 | . . . . . . . . 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 2744 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ℝ) = ((𝐹 ↾ ℝ) ↾ ℤ)) |
10 | 9 | reseq1d 5890 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)) = (((𝐹 ↾ ℝ) ↾ ℤ) ↾ (𝑀[,)+∞))) |
11 | resres 5904 | . . . . . . 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 43098 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) |
16 | 15 | eqcomd 2744 | . . . . . . 7 ⊢ (𝜑 → (ℤ ∩ (𝑀[,)+∞)) = 𝑍) |
17 | 16 | reseq2d 5891 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (ℤ ∩ (𝑀[,)+∞))) = ((𝐹 ↾ ℝ) ↾ 𝑍)) |
18 | 10, 12, 17 | 3eqtrrd 2783 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ 𝑍) = ((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) |
19 | 18 | fveq2d 6778 | . . . 4 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim sup‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞)))) |
20 | 13 | zred 12426 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
21 | eqid 2738 | . . . . 5 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
22 | limsupresuz.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
23 | 22 | resexd 5938 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ V) |
24 | 20, 21, 23 | limsupresico 43241 | . . . 4 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ (𝑀[,)+∞))) = (lim sup‘(𝐹 ↾ ℝ))) |
25 | 19, 24 | eqtrd 2778 | . . 3 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ ℝ) ↾ 𝑍)) = (lim sup‘(𝐹 ↾ ℝ))) |
26 | 3, 25 | eqtrd 2778 | . 2 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘(𝐹 ↾ ℝ))) |
27 | 22 | resexd 5938 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑍) ∈ V) |
28 | 27 | limsupresre 43237 | . 2 ⊢ (𝜑 → (lim sup‘((𝐹 ↾ 𝑍) ↾ ℝ)) = (lim sup‘(𝐹 ↾ 𝑍))) |
29 | 22 | limsupresre 43237 | . 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹)) |
30 | 26, 28, 29 | 3eqtr3d 2786 | 1 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 dom cdm 5589 ↾ cres 5591 Rel wrel 5594 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 +∞cpnf 11006 ℤcz 12319 ℤ≥cuz 12582 [,)cico 13081 lim supclsp 15179 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-ico 13085 df-limsup 15180 |
This theorem is referenced by: limsupresuz2 43250 |
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