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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 13470 | . . . 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 13466 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
6 | 3, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
7 | 3 | rexrd 11309 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 4 | rexrd 11309 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | icc0 13432 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) | |
10 | 7, 8, 9 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
11 | 10 | biimpar 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → (𝐴[,]𝐵) = ∅) |
12 | 11 | difeq1d 4135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (∅ ∖ (𝐴(,)𝐵))) |
13 | 0dif 4411 | . . . . . . 7 ⊢ (∅ ∖ (𝐴(,)𝐵)) = ∅ | |
14 | 0ss 4406 | . . . . . . 7 ⊢ ∅ ⊆ {𝐴, 𝐵} | |
15 | 13, 14 | eqsstri 4030 | . . . . . 6 ⊢ (∅ ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} |
16 | 12, 15 | eqsstrdi 4050 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 𝐴) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
17 | uncom 4168 | . . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) | |
18 | 7 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) |
19 | 8 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) |
20 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
21 | prunioo 13518 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
22 | 18, 19, 20, 21 | syl3anc 1370 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) |
23 | 17, 22 | eqtr2id 2788 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ({𝐴, 𝐵} ∪ (𝐴(,)𝐵))) |
24 | 23 | difeq1d 4135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵))) |
25 | difun2 4487 | . . . . . . 7 ⊢ (({𝐴, 𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) | |
26 | 24, 25 | eqtrdi 2791 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = ({𝐴, 𝐵} ∖ (𝐴(,)𝐵))) |
27 | difss 4146 | . . . . . 6 ⊢ ({𝐴, 𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵} | |
28 | 26, 27 | eqsstrdi 4050 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
29 | 16, 28, 4, 3 | ltlecasei 11367 | . . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) ⊆ {𝐴, 𝐵}) |
30 | 3, 4 | prssd 4827 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ ℝ) |
31 | prfi 9361 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin | |
32 | ovolfi 25543 | . . . . 5 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0) | |
33 | 31, 30, 32 | sylancr 587 | . . . 4 ⊢ (𝜑 → (vol*‘{𝐴, 𝐵}) = 0) |
34 | ovolssnul 25536 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∖ (𝐴(,)𝐵))) = 0) |
36 | itgioo.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ) | |
37 | 2, 6, 35, 36 | itgss3 25865 | . 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1) ∧ ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)) |
38 | 37 | simprd 495 | 1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 {cpr 4633 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 [,]cicc 13387 vol*covol 25511 𝐿1cibl 25666 ∫citg 25667 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 |
This theorem is referenced by: itgpowd 26106 lcmineqlem10 42020 lcmineqlem12 42022 itgioocnicc 45933 itgiccshift 45936 itgperiod 45937 fourierdlem73 46135 fourierdlem81 46143 fourierdlem82 46144 fourierdlem111 46173 |
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