Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sublevolico | Structured version Visualization version GIF version |
Description: The Lebesgue measure of a left-closed, right-open interval is greater than or equal to the difference of the two bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
sublevolico.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
sublevolico.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
sublevolico | ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sublevolico.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | sublevolico.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 1, 2 | resubcld 11070 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
4 | eqidd 2824 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐵 − 𝐴)) | |
5 | 3, 4 | eqled 10745 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴)) |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴)) |
7 | volico 42275 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) | |
8 | 2, 1, 7 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
10 | iftrue 4475 | . . . . 5 ⊢ (𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴)) | |
11 | 10 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = (𝐵 − 𝐴)) |
12 | 9, 11 | eqtr2d 2859 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) = (vol‘(𝐴[,)𝐵))) |
13 | 6, 12 | breqtrd 5094 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵))) |
14 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → ¬ 𝐴 < 𝐵) | |
15 | 1, 2 | lenltd 10788 | . . . . . 6 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
16 | 15 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
17 | 14, 16 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → 𝐵 ≤ 𝐴) |
18 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) |
19 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) |
20 | 18, 19 | suble0d 11233 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → ((𝐵 − 𝐴) ≤ 0 ↔ 𝐵 ≤ 𝐴)) |
21 | 17, 20 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → (𝐵 − 𝐴) ≤ 0) |
22 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0)) |
23 | iffalse 4478 | . . . . 5 ⊢ (¬ 𝐴 < 𝐵 → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0) | |
24 | 23 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → if(𝐴 < 𝐵, (𝐵 − 𝐴), 0) = 0) |
25 | 22, 24 | eqtr2d 2859 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → 0 = (vol‘(𝐴[,)𝐵))) |
26 | 21, 25 | breqtrd 5094 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝐵) → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵))) |
27 | 13, 26 | pm2.61dan 811 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 < clt 10677 ≤ cle 10678 − cmin 10872 [,)cico 12743 volcvol 24066 |
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 |
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-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-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-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-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-rlim 14848 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 |
This theorem is referenced by: ovolval5lem1 42941 |
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