![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10127 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | itgspliticc.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶)) | |
4 | itgspliticc.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | elicc2 12276 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))) | |
6 | 1, 4, 5 | syl2anc 694 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))) |
7 | 3, 6 | mpbid 222 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) |
8 | 7 | simp1d 1093 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
9 | 8 | rexrd 10127 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
10 | 4 | rexrd 10127 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
11 | df-icc 12220 | . . . . . . 7 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
12 | xrmaxle 12052 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝑧 ↔ (𝐴 ≤ 𝑧 ∧ 𝐵 ≤ 𝑧))) | |
13 | xrlemin 12053 | . . . . . . 7 ⊢ ((𝑧 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑧 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝑧 ≤ 𝐵 ∧ 𝑧 ≤ 𝐶))) | |
14 | 11, 12, 13 | ixxin 12230 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = (if(𝐴 ≤ 𝐵, 𝐵, 𝐴)[,]if(𝐵 ≤ 𝐶, 𝐵, 𝐶))) |
16 | 7 | simp2d 1094 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
17 | 16 | iftrued 4127 | . . . . . 6 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐵) |
18 | 7 | simp3d 1095 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
19 | 18 | iftrued 4127 | . . . . . 6 ⊢ (𝜑 → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) = 𝐵) |
20 | 17, 19 | oveq12d 6708 | . . . . 5 ⊢ (𝜑 → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴)[,]if(𝐵 ≤ 𝐶, 𝐵, 𝐶)) = (𝐵[,]𝐵)) |
21 | iccid 12258 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → (𝐵[,]𝐵) = {𝐵}) | |
22 | 9, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵[,]𝐵) = {𝐵}) |
23 | 15, 20, 22 | 3eqtrd 2689 | . . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵}) |
24 | 23 | fveq2d 6233 | . . 3 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∩ (𝐵[,]𝐶))) = (vol*‘{𝐵})) |
25 | ovolsn 23309 | . . . 4 ⊢ (𝐵 ∈ ℝ → (vol*‘{𝐵}) = 0) | |
26 | 8, 25 | syl 17 | . . 3 ⊢ (𝜑 → (vol*‘{𝐵}) = 0) |
27 | 24, 26 | eqtrd 2685 | . 2 ⊢ (𝜑 → (vol*‘((𝐴[,]𝐵) ∩ (𝐵[,]𝐶))) = 0) |
28 | iccsplit 12343 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))) | |
29 | 1, 4, 3, 28 | syl3anc 1366 | . 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 23647 | 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 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∪ cun 3605 ∩ cin 3606 ifcif 4119 {csn 4210 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 + caddc 9977 ℝ*cxr 10111 ≤ cle 10113 [,]cicc 12216 vol*covol 23277 𝐿1cibl 23431 ∫citg 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-ofr 6940 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461 df-rest 16130 df-topgen 16151 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-top 20747 df-topon 20764 df-bases 20798 df-cmp 21238 df-ovol 23279 df-vol 23280 df-mbf 23433 df-itg1 23434 df-itg2 23435 df-ibl 23436 df-itg 23437 |
This theorem is referenced by: itgspltprt 40513 fourierdlem107 40748 |
Copyright terms: Public domain | W3C validator |