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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 38849 | . 2 ⊢ ∅ ∈ MblFn | |
2 | fconstmpt 5085 | . . . . . . 7 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
3 | 2 | eqcomi 2619 | . . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ 0) = (ℝ × {0}) |
4 | 3 | fveq2i 6106 | . . . . 5 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = (∫2‘(ℝ × {0})) |
5 | itg20 23310 | . . . . 5 ⊢ (∫2‘(ℝ × {0})) = 0 | |
6 | 4, 5 | eqtri 2632 | . . . 4 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) = 0 |
7 | 0re 9919 | . . . 4 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | eqeltri 2684 | . . 3 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
9 | 8 | rgenw 2908 | . 2 ⊢ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ |
10 | noel 3878 | . . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅ | |
11 | 10 | intnanr 952 | . . . . . . . 8 ⊢ ¬ (𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))) |
12 | 11 | iffalsei 4046 | . . . . . . 7 ⊢ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0) = 0 |
13 | 12 | eqcomi 2619 | . . . . . 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 4672 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ 0) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ∅ ∧ 0 ≤ (ℜ‘(0 / (i↑𝑘)))), (ℜ‘(0 / (i↑𝑘))), 0))) |
16 | eqidd 2611 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (ℜ‘(0 / (i↑𝑘))) = (ℜ‘(0 / (i↑𝑘)))) | |
17 | dm0 5260 | . . . . 5 ⊢ dom ∅ = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → dom ∅ = ∅) |
19 | 10 | intnan 951 | . . . . 5 ⊢ ¬ (⊤ ∧ 𝑥 ∈ ∅) |
20 | 19 | pm2.21i 115 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → (∅‘𝑥) = 0) |
21 | 15, 16, 18, 20 | isibl 23338 | . . 3 ⊢ (⊤ → (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ))) |
22 | 21 | trud 1484 | . 2 ⊢ (∅ ∈ 𝐿1 ↔ (∅ ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ 0)) ∈ ℝ)) |
23 | 1, 9, 22 | mpbir2an 957 | 1 ⊢ ∅ ∈ 𝐿1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ici 9817 ≤ cle 9954 / cdiv 10563 3c3 10948 ...cfz 12197 ↑cexp 12722 ℜcre 13685 MblFncmbf 23189 ∫2citg2 23191 𝐿1cibl 23192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-itg2 23196 df-ibl 23197 df-0p 23243 |
This theorem is referenced by: itgvol0 38860 |
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