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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iblempty | Structured version Visualization version GIF version |
Description: The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iblempty | ⊢ ∅ ∈ 𝐿1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbf0 24998 | . 2 ⊢ ∅ ∈ MblFn | |
2 | fconstmpt 5694 | . . . . . . 7 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
3 | 2 | eqcomi 2745 | . . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ 0) = (ℝ × {0}) |
4 | 3 | fveq2i 6845 | . . . . 5 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = (∫2‘(ℝ × {0})) |
5 | itg20 25102 | . . . . 5 ⊢ (∫2‘(ℝ × {0})) = 0 | |
6 | 4, 5 | eqtri 2764 | . . . 4 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = 0 |
7 | 0re 11157 | . . . 4 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | eqeltri 2834 | . . 3 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
9 | 8 | rgenw 3068 | . 2 ⊢ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
10 | noel 4290 | . . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅ | |
11 | 10 | intnanr 488 | . . . . . . . 8 ⊢ ¬ (𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))) |
12 | 11 | iffalsei 4496 | . . . . . . 7 ⊢ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0 |
13 | 12 | eqcomi 2745 | . . . . . 6 ⊢ 0 = if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 = if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0)) |
15 | 14 | mpteq2dva 5205 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ 0) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) |
16 | eqidd 2737 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘)))) | |
17 | dm0 5876 | . . . . 5 ⊢ dom ∅ = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → dom ∅ = ∅) |
19 | 10 | intnan 487 | . . . . 5 ⊢ ¬ (⊤ ∧ 𝑥 ∈ ∅) |
20 | 19 | pm2.21i 119 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (∅‘𝑥) = 0) |
21 | 15, 16, 18, 20 | isibl 25130 | . . 3 ⊢ (⊤ → (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ))) |
22 | 21 | mptru 1548 | . 2 ⊢ (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ)) |
23 | 1, 9, 22 | mpbir2an 709 | 1 ⊢ ∅ ∈ 𝐿1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ∀wral 3064 ∅c0 4282 ifcif 4486 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 dom cdm 5633 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 ici 11053 ≤ cle 11190 / cdiv 11812 3c3 12209 ...cfz 13424 ↑cexp 13967 ℜcre 14982 MblFncmbf 24978 ∫2citg2 24980 𝐿1cibl 24981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xadd 13034 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-xmet 20789 df-met 20790 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-0p 25034 |
This theorem is referenced by: itgvol0 44199 |
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