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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 12821 | . . . 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 12817 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
6 | 3, 4, 5 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
7 | 3 | rexrd 10690 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 4 | rexrd 10690 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | icc0 12785 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) | |
10 | 7, 8, 9 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
11 | 10 | biimpar 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅) |
12 | 11 | difeq1d 4097 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵))) |
13 | 0dif 4354 | . . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅ | |
14 | 0ss 4349 | . . . . . . 7 ⊢ ∅ ⊆ {𝐴, 𝐵} | |
15 | 13, 14 | eqsstri 4000 | . . . . . 6 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} |
16 | 12, 15 | eqsstrdi 4020 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
17 | uncom 4128 | . . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) | |
18 | 7 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) |
19 | 8 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) |
20 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
21 | prunioo 12866 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
22 | 18, 19, 20, 21 | syl3anc 1367 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) |
23 | 17, 22 | syl5req 2869 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ({𝐴, 𝐵} ∪ (𝐴(,)𝐵))) |
24 | 23 | difeq1d 4097 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵))) |
25 | difun2 4428 | . . . . . . 7 ⊢ (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) | |
26 | 24, 25 | syl6eq 2872 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵))) |
27 | difss 4107 | . . . . . 6 ⊢ ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} | |
28 | 26, 27 | eqsstrdi 4020 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
29 | 16, 28, 4, 3 | ltlecasei 10747 | . . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
30 | 3, 4 | prssd 4754 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ) |
31 | prfi 8792 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin | |
32 | ovolfi 24094 | . . . . 5 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0) | |
33 | 31, 30, 32 | sylancr 589 | . . . 4 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0) |
34 | ovolssnul 24087 | . . . 4 ⊢ ((((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ⊆ ℝ ∧ (vol*‘{𝐴, 𝐵}) = 0) → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) | |
35 | 29, 30, 33, 34 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) |
36 | itgioo.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) | |
37 | 2, 6, 35, 36 | itgss3 24414 | . 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1) ∧ ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)) |
38 | 37 | simprd 498 | 1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {cpr 4568 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 ℂcc 10534 ℝcr 10535 0cc0 10536 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 (,)cioo 12737 [,]cicc 12740 vol*covol 24062 𝐿1cibl 24217 ∫citg 24218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-symdif 4218 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-rest 16695 df-topgen 16716 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-top 21501 df-topon 21518 df-bases 21553 df-cmp 21994 df-ovol 24064 df-vol 24065 df-mbf 24219 df-itg1 24220 df-itg2 24221 df-ibl 24222 df-itg 24223 |
This theorem is referenced by: itgpowd 39819 itgioocnicc 42260 itgiccshift 42263 itgperiod 42264 fourierdlem73 42463 fourierdlem81 42471 fourierdlem82 42472 fourierdlem111 42501 |
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