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Mirrors > Home > MPE Home > Th. List > itgioo | Structured version Visualization version GIF version |
Description: Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.) |
Ref | Expression |
---|---|
itgioo.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
itgioo.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
itgioo.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
itgioo | ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossicc 12571 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
3 | itgioo.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | itgioo.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | iccssre 12567 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
6 | 3, 4, 5 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
7 | 3 | rexrd 10426 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 4 | rexrd 10426 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | icc0 12535 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) | |
10 | 7, 8, 9 | syl2anc 579 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
11 | 10 | biimpar 471 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅) |
12 | 11 | difeq1d 3949 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵))) |
13 | 0dif 4202 | . . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅ | |
14 | 0ss 4197 | . . . . . . 7 ⊢ ∅ ⊆ {𝐴, 𝐵} | |
15 | 13, 14 | eqsstri 3853 | . . . . . 6 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} |
16 | 12, 15 | syl6eqss 3873 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
17 | uncom 3979 | . . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) | |
18 | 7 | adantr 474 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) |
19 | 8 | adantr 474 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) |
20 | simpr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
21 | prunioo 12618 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
22 | 18, 19, 20, 21 | syl3anc 1439 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) |
23 | 17, 22 | syl5req 2826 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ({𝐴, 𝐵} ∪ (𝐴(,)𝐵))) |
24 | 23 | difeq1d 3949 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵))) |
25 | difun2 4271 | . . . . . . 7 ⊢ (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) | |
26 | 24, 25 | syl6eq 2829 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵))) |
27 | difss 3959 | . . . . . 6 ⊢ ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} | |
28 | 26, 27 | syl6eqss 3873 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
29 | 16, 28, 4, 3 | ltlecasei 10484 | . . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
30 | prssi 4583 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) | |
31 | 3, 4, 30 | syl2anc 579 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ) |
32 | prfi 8523 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin | |
33 | ovolfi 23698 | . . . . 5 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0) | |
34 | 32, 31, 33 | sylancr 581 | . . . 4 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0) |
35 | ovolssnul 23691 | . . . 4 ⊢ ((((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ⊆ ℝ ∧ (vol*‘{𝐴, 𝐵}) = 0) → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) | |
36 | 29, 31, 34, 35 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) |
37 | itgioo.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) | |
38 | 2, 6, 36, 37 | itgss3 24018 | . 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1) ∧ ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)) |
39 | 38 | simprd 491 | 1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∖ cdif 3788 ∪ cun 3789 ⊆ wss 3791 ∅c0 4140 {cpr 4399 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 ℂcc 10270 ℝcr 10271 0cc0 10272 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 (,)cioo 12487 [,]cicc 12490 vol*covol 23666 𝐿1cibl 23821 ∫citg 23822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-symdif 4066 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-rest 16469 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-bases 21158 df-cmp 21599 df-ovol 23668 df-vol 23669 df-mbf 23823 df-itg1 23824 df-itg2 23825 df-ibl 23826 df-itg 23827 |
This theorem is referenced by: itgpowd 38751 itgioocnicc 41113 itgiccshift 41116 itgperiod 41117 fourierdlem73 41316 fourierdlem81 41324 fourierdlem82 41325 fourierdlem111 41354 |
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