Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > itgvol0 | Structured version Visualization version GIF version |
Description: If the domani is negligible, the function is integrable and the integral is 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
itgvol0.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
itgvol0.2 | ⊢ (𝜑 → (vol*‘𝐴) = 0) |
itgvol0.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
itgvol0 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ∧ ∫𝐴𝐵 d𝑥 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt0 6480 | . . . 4 ⊢ (𝑥 ∈ ∅ ↦ 𝐵) = ∅ | |
2 | iblempty 43071 | . . . 4 ⊢ ∅ ∈ 𝐿1 | |
3 | 1, 2 | eqeltri 2830 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ 𝐵) ∈ 𝐿1 |
4 | 0ss 4286 | . . . . . 6 ⊢ ∅ ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∅ ⊆ 𝐴) |
6 | itgvol0.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
7 | difssd 4024 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ ∅) ⊆ 𝐴) | |
8 | itgvol0.2 | . . . . . 6 ⊢ (𝜑 → (vol*‘𝐴) = 0) | |
9 | ovolssnul 24242 | . . . . . 6 ⊢ (((𝐴 ∖ ∅) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝐴 ∖ ∅)) = 0) | |
10 | 7, 6, 8, 9 | syl3anc 1372 | . . . . 5 ⊢ (𝜑 → (vol*‘(𝐴 ∖ ∅)) = 0) |
11 | itgvol0.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
12 | 5, 6, 10, 11 | itgss3 24570 | . . . 4 ⊢ (𝜑 → (((𝑥 ∈ ∅ ↦ 𝐵) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) ∧ ∫∅𝐵 d𝑥 = ∫𝐴𝐵 d𝑥)) |
13 | 12 | simpld 498 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ∅ ↦ 𝐵) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)) |
14 | 3, 13 | mpbii 236 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
15 | 12 | simprd 499 | . . 3 ⊢ (𝜑 → ∫∅𝐵 d𝑥 = ∫𝐴𝐵 d𝑥) |
16 | itg0 24535 | . . 3 ⊢ ∫∅𝐵 d𝑥 = 0 | |
17 | 15, 16 | eqtr3di 2789 | . 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = 0) |
18 | 14, 17 | jca 515 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ∧ ∫𝐴𝐵 d𝑥 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3841 ⊆ wss 3844 ∅c0 4212 ↦ cmpt 5111 ‘cfv 6340 ℂcc 10616 ℝcr 10617 0cc0 10618 vol*covol 24217 𝐿1cibl 24372 ∫citg 24373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-symdif 4134 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-disj 4997 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-ofr 7429 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-er 8323 df-map 8442 df-pm 8443 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fi 8951 df-sup 8982 df-inf 8983 df-oi 9050 df-dju 9406 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-n0 11980 df-z 12066 df-uz 12328 df-q 12434 df-rp 12476 df-xneg 12593 df-xadd 12594 df-xmul 12595 df-ioo 12828 df-ico 12830 df-icc 12831 df-fz 12985 df-fzo 13128 df-fl 13256 df-mod 13332 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-sum 15139 df-rest 16802 df-topgen 16823 df-psmet 20212 df-xmet 20213 df-met 20214 df-bl 20215 df-mopn 20216 df-top 21648 df-topon 21665 df-bases 21700 df-cmp 22141 df-ovol 24219 df-vol 24220 df-mbf 24374 df-itg1 24375 df-itg2 24376 df-ibl 24377 df-itg 24378 df-0p 24425 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |