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Mirrors > Home > MPE Home > Th. List > iccmbl | Structured version Visualization version GIF version |
Description: A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
iccmbl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssre 12543 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
2 | dfss4 4088 | . . 3 ⊢ ((𝐴[,]𝐵) ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ (𝐴[,]𝐵))) = (𝐴[,]𝐵)) | |
3 | 1, 2 | sylib 210 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (ℝ ∖ (𝐴[,]𝐵))) = (𝐴[,]𝐵)) |
4 | difreicc 12597 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) = ((-∞(,)𝐴) ∪ (𝐵(,)+∞))) | |
5 | ioombl 23731 | . . . . 5 ⊢ (-∞(,)𝐴) ∈ dom vol | |
6 | ioombl 23731 | . . . . 5 ⊢ (𝐵(,)+∞) ∈ dom vol | |
7 | unmbl 23703 | . . . . 5 ⊢ (((-∞(,)𝐴) ∈ dom vol ∧ (𝐵(,)+∞) ∈ dom vol) → ((-∞(,)𝐴) ∪ (𝐵(,)+∞)) ∈ dom vol) | |
8 | 5, 6, 7 | mp2an 683 | . . . 4 ⊢ ((-∞(,)𝐴) ∪ (𝐵(,)+∞)) ∈ dom vol |
9 | 4, 8 | syl6eqel 2914 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) ∈ dom vol) |
10 | cmmbl 23700 | . . 3 ⊢ ((ℝ ∖ (𝐴[,]𝐵)) ∈ dom vol → (ℝ ∖ (ℝ ∖ (𝐴[,]𝐵))) ∈ dom vol) | |
11 | 9, 10 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (ℝ ∖ (𝐴[,]𝐵))) ∈ dom vol) |
12 | 3, 11 | eqeltrrd 2907 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∖ cdif 3795 ∪ cun 3796 ⊆ wss 3798 dom cdm 5342 (class class class)co 6905 ℝcr 10251 +∞cpnf 10388 -∞cmnf 10389 (,)cioo 12463 [,]cicc 12466 volcvol 23629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xadd 12233 df-ioo 12467 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-sum 14794 df-xmet 20099 df-met 20100 df-ovol 23630 df-vol 23631 |
This theorem is referenced by: iccvolcl 23733 ovolioo 23734 dyadmbl 23766 volsup2 23771 volcn 23772 volivth 23773 mbfi1fseqlem4 23884 cniccibl 24006 ftc1lem4 24201 cnicciblnc 34017 ftc1cnnclem 34019 areacirc 34041 iocmbl 38633 arearect 38636 areaquad 38637 volicc 41002 fourierdlem87 41197 fourierdlem107 41217 |
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