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Mirrors > Home > MPE Home > Th. List > ovolicc | Structured version Visualization version GIF version |
Description: The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
ovolicc | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssre 12632 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
2 | 1 | 3adant3 1112 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) ⊆ ℝ) |
3 | ovolcl 23794 | . . 3 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol*‘(𝐴[,]𝐵)) ∈ ℝ*) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) ∈ ℝ*) |
5 | simp2 1117 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) | |
6 | simp1 1116 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) | |
7 | 5, 6 | resubcld 10867 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) ∈ ℝ) |
8 | 7 | rexrd 10488 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) ∈ ℝ*) |
9 | simp3 1118 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
10 | eqeq1 2776 | . . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 = 1 ↔ 𝑛 = 1)) | |
11 | 10 | ifbid 4366 | . . . 4 ⊢ (𝑚 = 𝑛 → if(𝑚 = 1, 〈𝐴, 𝐵〉, 〈0, 0〉) = if(𝑛 = 1, 〈𝐴, 𝐵〉, 〈0, 0〉)) |
12 | 11 | cbvmptv 5024 | . . 3 ⊢ (𝑚 ∈ ℕ ↦ if(𝑚 = 1, 〈𝐴, 𝐵〉, 〈0, 0〉)) = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 〈𝐴, 𝐵〉, 〈0, 0〉)) |
13 | 6, 5, 9, 12 | ovolicc1 23832 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴)) |
14 | eqeq1 2776 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))) | |
15 | 14 | anbi2d 619 | . . . . 5 ⊢ (𝑧 = 𝑦 → (((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
16 | 15 | rexbidv 3236 | . . . 4 ⊢ (𝑧 = 𝑦 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
17 | 16 | cbvrabv 3406 | . . 3 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} |
18 | 6, 5, 9, 17 | ovolicc2 23838 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵))) |
19 | 4, 8, 13, 18 | xrletrid 12363 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∃wrex 3083 {crab 3086 ∩ cin 3822 ⊆ wss 3823 ifcif 4344 〈cop 4441 ∪ cuni 4708 class class class wbr 4925 ↦ cmpt 5004 × cxp 5401 ran crn 5404 ∘ ccom 5407 ‘cfv 6185 (class class class)co 6974 ↑𝑚 cmap 8204 supcsup 8697 ℝcr 10332 0cc0 10333 1c1 10334 + caddc 10336 ℝ*cxr 10471 < clt 10472 ≤ cle 10473 − cmin 10668 ℕcn 11437 (,)cioo 12552 [,]cicc 12555 seqcseq 13182 abscabs 14452 vol*covol 23778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-sum 14902 df-rest 16550 df-topgen 16571 df-psmet 20251 df-xmet 20252 df-met 20253 df-bl 20254 df-mopn 20255 df-top 21218 df-topon 21235 df-bases 21270 df-cmp 21711 df-ovol 23780 |
This theorem is referenced by: ovolicopnf 23840 iccvolcl 23883 ovolioo 23884 dyadovol 23909 volcn 23922 vitalilem4 23927 vitalilem5 23928 ftc1a 24349 areacirc 34457 arearect 39247 areaquad 39248 volicc 41739 |
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