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Mirrors > Home > MPE Home > Th. List > itgspliticc | Structured version Visualization version GIF version |
Description: The ∫ integral splits on closed intervals with matching endpoints. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
itgspliticc.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
itgspliticc.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
itgspliticc.3 | ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶)) |
itgspliticc.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐶)) → 𝐷 ∈ 𝑉) |
itgspliticc.5 | ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐷) ∈ 𝐿1) |
itgspliticc.6 | ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↦ 𝐷) ∈ 𝐿1) |
Ref | Expression |
---|---|
itgspliticc | ⊢ (𝜑 → ∫(𝐴[,]𝐶)𝐷 d𝑥 = (∫(𝐴[,]𝐵)𝐷 d𝑥 + ∫(𝐵[,]𝐶)𝐷 d𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgspliticc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 10693 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | itgspliticc.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶)) | |
4 | itgspliticc.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | elicc2 12804 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))) | |
6 | 1, 4, 5 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))) |
7 | 3, 6 | mpbid 234 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) |
8 | 7 | simp1d 1138 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
9 | 8 | rexrd 10693 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
10 | 4 | rexrd 10693 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
11 | df-icc 12748 | . . . . . . 7 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
12 | xrmaxle 12579 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝑧 ↔ (𝐴 ≤ 𝑧 ∧ 𝐵 ≤ 𝑧))) | |
13 | xrlemin 12580 | . . . . . . 7 ⊢ ((𝑧 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑧 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝑧 ≤ 𝐵 ∧ 𝑧 ≤ 𝐶))) | |
14 | 11, 12, 13 | ixxin 12758 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = (if(𝐴 ≤ 𝐵, 𝐵, 𝐴)[,]if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) |
15 | 2, 9, 9, 10, 14 | syl22anc 836 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = (if(𝐴 ≤ 𝐵, 𝐵, 𝐴)[,]if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) |
16 | 7 | simp2d 1139 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
17 | 16 | iftrued 4477 | . . . . . 6 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵) |
18 | 7 | simp3d 1140 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
19 | 18 | iftrued 4477 | . . . . . 6 ⊢ (𝜑 → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) = 𝐵) |
20 | 17, 19 | oveq12d 7176 | . . . . 5 ⊢ (𝜑 → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴)[,]if(𝐵 ≤ 𝐶, 𝐵, 𝐶)) = (𝐵[,]𝐵)) |
21 | iccid 12786 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → (𝐵[,]𝐵) = {𝐵}) | |
22 | 9, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵[,]𝐵) = {𝐵}) |
23 | 15, 20, 22 | 3eqtrd 2862 | . . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵}) |
24 | 23 | fveq2d 6676 | . . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∩ (𝐵[,]𝐶))) = (vol*‘{𝐵})) |
25 | ovolsn 24098 | . . . 4 ⊢ (𝐵 ∈ ℝ → (vol*‘{𝐵}) = 0) | |
26 | 8, 25 | syl 17 | . . 3 ⊢ (𝜑 → (vol*‘{𝐵}) = 0) |
27 | 24, 26 | eqtrd 2858 | . 2 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∩ (𝐵[,]𝐶))) = 0) |
28 | iccsplit 12874 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) | |
29 | 1, 4, 3, 28 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) |
30 | itgspliticc.4 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐶)) → 𝐷 ∈ 𝑉) | |
31 | itgspliticc.5 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐷) ∈ 𝐿1) | |
32 | itgspliticc.6 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↦ 𝐷) ∈ 𝐿1) | |
33 | 27, 29, 30, 31, 32 | itgsplit 24438 | 1 ⊢ (𝜑 → ∫(𝐴[,]𝐶)𝐷 d𝑥 = (∫(𝐴[,]𝐵)𝐷 d𝑥 + ∫(𝐵[,]𝐶)𝐷 d𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ifcif 4469 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 ℝ*cxr 10676 ≤ cle 10678 [,]cicc 12744 vol*covol 24065 𝐿1cibl 24220 ∫citg 24221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-rest 16698 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-cmp 21997 df-ovol 24067 df-vol 24068 df-mbf 24222 df-itg1 24223 df-itg2 24224 df-ibl 24225 df-itg 24226 |
This theorem is referenced by: itgspltprt 42271 fourierdlem107 42505 |
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