Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ibliooicc | Structured version Visualization version GIF version |
Description: If a function is integrable on an open interval, it is integrable on the corresponding closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ibliooicc.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ibliooicc.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ibliooicc.3 | ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1) |
ibliooicc.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
ibliooicc | ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibliooicc.3 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1) | |
2 | ioossicc 12919 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
4 | ibliooicc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | ibliooicc.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 4, 5 | iccssred 12920 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
7 | 4 | rexrd 10781 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 5 | rexrd 10781 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | icc0 12881 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) | |
10 | 7, 8, 9 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
11 | 10 | biimpar 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅) |
12 | 11 | difeq1d 4022 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵))) |
13 | 0dif 4300 | . . . . . . . 8 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅ | |
14 | 0ss 4295 | . . . . . . . 8 ⊢ ∅ ⊆ {𝐴, 𝐵} | |
15 | 13, 14 | eqsstri 3921 | . . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} |
16 | 12, 15 | eqsstrdi 3941 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
17 | ssid 3909 | . . . . . . 7 ⊢ ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) | |
18 | 7 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) |
19 | 8 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) |
20 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
21 | iccdifioo 42633 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵}) | |
22 | 18, 19, 20, 21 | syl3anc 1372 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵}) |
23 | 17, 22 | sseqtrid 3939 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
24 | 16, 23, 5, 4 | ltlecasei 10838 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
25 | prssi 4719 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
26 | 4, 5, 25 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ) |
27 | prfi 8879 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin | |
28 | ovolfi 24258 | . . . . . 6 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0) | |
29 | 27, 26, 28 | sylancr 590 | . . . . 5 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0) |
30 | ovolssnul 24251 | . . . . 5 ⊢ ((((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ⊆ ℝ ∧ (vol*‘{𝐴, 𝐵}) = 0) → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) | |
31 | 24, 26, 29, 30 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) |
32 | ibliooicc.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) | |
33 | 3, 6, 31, 32 | itgss3 24579 | . . 3 ⊢ (𝜑 → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1) ∧ ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)) |
34 | 33 | simpld 498 | . 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1)) |
35 | 1, 34 | mpbid 235 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3850 ⊆ wss 3853 ∅c0 4221 {cpr 4528 class class class wbr 5040 ↦ cmpt 5120 ‘cfv 6349 (class class class)co 7182 Fincfn 8567 ℂcc 10625 ℝcr 10626 0cc0 10627 ℝ*cxr 10764 < clt 10765 ≤ cle 10766 (,)cioo 12833 [,]cicc 12836 vol*covol 24226 𝐿1cibl 24381 ∫citg 24382 |
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 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 |
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 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-symdif 4143 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-disj 5006 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-ofr 7438 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-er 8332 df-map 8451 df-pm 8452 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-dju 9415 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-n0 11989 df-z 12075 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-ico 12839 df-icc 12840 df-fz 12994 df-fzo 13137 df-fl 13265 df-mod 13341 df-seq 13473 df-exp 13534 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-sum 15148 df-rest 16811 df-topgen 16832 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-top 21657 df-topon 21674 df-bases 21709 df-cmp 22150 df-ovol 24228 df-vol 24229 df-mbf 24383 df-itg1 24384 df-itg2 24385 df-ibl 24386 df-itg 24387 |
This theorem is referenced by: fourierdlem69 43298 fourierdlem73 43302 fourierdlem81 43310 fourierdlem93 43322 |
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