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Mathbox for Glauco Siliprandi |
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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 6647 | . . . 4 ⊢ (𝑥 ∈ ∅ ↦ 𝐵) = ∅ | |
2 | iblempty 44296 | . . . 4 ⊢ ∅ ∈ 𝐿1 | |
3 | 1, 2 | eqeltri 2830 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ 𝐵) ∈ 𝐿1 |
4 | 0ss 4360 | . . . . . 6 ⊢ ∅ ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∅ ⊆ 𝐴) |
6 | itgvol0.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
7 | difssd 4096 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∖ ∅) ⊆ 𝐴) | |
8 | itgvol0.2 | . . . . . 6 ⊢ (𝜑 → (vol*‘𝐴) = 0) | |
9 | ovolssnul 24874 | . . . . . 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 25202 | . . . 4 ⊢ (𝜑 → (((𝑥 ∈ ∅ ↦ 𝐵) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) ∧ ∫∅𝐵 d𝑥 = ∫𝐴𝐵 d𝑥)) |
13 | 12 | simpld 496 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ∅ ↦ 𝐵) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)) |
14 | 3, 13 | mpbii 232 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
15 | 12 | simprd 497 | . . 3 ⊢ (𝜑 → ∫∅𝐵 d𝑥 = ∫𝐴𝐵 d𝑥) |
16 | itg0 25167 | . . 3 ⊢ ∫∅𝐵 d𝑥 = 0 | |
17 | 15, 16 | eqtr3di 2788 | . 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = 0) |
18 | 14, 17 | jca 513 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ∧ ∫𝐴𝐵 d𝑥 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3911 ⊆ wss 3914 ∅c0 4286 ↦ cmpt 5192 ‘cfv 6500 ℂcc 11057 ℝcr 11058 0cc0 11059 vol*covol 24849 𝐿1cibl 25004 ∫citg 25005 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-symdif 4206 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-disj 5075 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-sum 15580 df-rest 17312 df-topgen 17333 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-bases 22319 df-cmp 22761 df-ovol 24851 df-vol 24852 df-mbf 25006 df-itg1 25007 df-itg2 25008 df-ibl 25009 df-itg 25010 df-0p 25057 |
This theorem is referenced by: (None) |
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