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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 23954 | . 2 ⊢ ∅ ∈ MblFn | |
2 | fconstmpt 5461 | . . . . . . 7 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
3 | 2 | eqcomi 2782 | . . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ 0) = (ℝ × {0}) |
4 | 3 | fveq2i 6500 | . . . . 5 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = (∫2‘(ℝ × {0})) |
5 | itg20 24057 | . . . . 5 ⊢ (∫2‘(ℝ × {0})) = 0 | |
6 | 4, 5 | eqtri 2797 | . . . 4 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = 0 |
7 | 0re 10440 | . . . 4 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | eqeltri 2857 | . . 3 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
9 | 8 | rgenw 3095 | . 2 ⊢ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
10 | noel 4178 | . . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅ | |
11 | 10 | intnanr 480 | . . . . . . . 8 ⊢ ¬ (𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))) |
12 | 11 | iffalsei 4355 | . . . . . . 7 ⊢ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0 |
13 | 12 | eqcomi 2782 | . . . . . 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 5019 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ 0) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) |
16 | eqidd 2774 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘)))) | |
17 | dm0 5635 | . . . . 5 ⊢ dom ∅ = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → dom ∅ = ∅) |
19 | 10 | intnan 479 | . . . . 5 ⊢ ¬ (⊤ ∧ 𝑥 ∈ ∅) |
20 | 19 | pm2.21i 117 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (∅‘𝑥) = 0) |
21 | 15, 16, 18, 20 | isibl 24085 | . . 3 ⊢ (⊤ → (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ))) |
22 | 21 | mptru 1515 | . 2 ⊢ (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ)) |
23 | 1, 9, 22 | mpbir2an 699 | 1 ⊢ ∅ ∈ 𝐿1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1508 ⊤wtru 1509 ∈ wcel 2051 ∀wral 3083 ∅c0 4173 ifcif 4345 {csn 4436 class class class wbr 4926 ↦ cmpt 5005 × cxp 5402 dom cdm 5404 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 0cc0 10334 ici 10336 ≤ cle 10474 / cdiv 11097 3c3 11495 ...cfz 12707 ↑cexp 13243 ℜcre 14316 MblFncmbf 23934 ∫2citg2 23936 𝐿1cibl 23937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 ax-addf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-disj 4895 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-ofr 7227 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-sup 8700 df-inf 8701 df-oi 8768 df-dju 9123 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-z 11793 df-uz 12058 df-q 12162 df-rp 12204 df-xadd 12324 df-ioo 12557 df-ico 12559 df-icc 12560 df-fz 12708 df-fzo 12849 df-fl 12976 df-seq 13184 df-exp 13244 df-hash 13505 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-clim 14705 df-sum 14903 df-xmet 20256 df-met 20257 df-ovol 23784 df-vol 23785 df-mbf 23939 df-itg1 23940 df-itg2 23941 df-ibl 23942 df-0p 23990 |
This theorem is referenced by: itgvol0 41713 |
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