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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoimbllem | Structured version Visualization version GIF version |
Description: Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hoimbllem.x | β’ (π β π β Fin) |
hoimbllem.n | β’ (π β π β β ) |
hoimbllem.s | β’ π = dom (volnβπ) |
hoimbllem.a | β’ (π β π΄:πβΆβ) |
hoimbllem.b | β’ (π β π΅:πβΆβ) |
hoimbllem.h | β’ π» = (π₯ β Fin β¦ (π β π₯, π¦ β β β¦ Xπ β π₯ if(π = π, (-β(,)π¦), β))) |
Ref | Expression |
---|---|
hoimbllem | β’ (π β Xπ β π ((π΄βπ)[,)(π΅βπ)) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoimbllem.x | . . 3 β’ (π β π β Fin) | |
2 | hoimbllem.n | . . 3 β’ (π β π β β ) | |
3 | hoimbllem.a | . . 3 β’ (π β π΄:πβΆβ) | |
4 | hoimbllem.b | . . 3 β’ (π β π΅:πβΆβ) | |
5 | hoimbllem.h | . . 3 β’ π» = (π₯ β Fin β¦ (π β π₯, π¦ β β β¦ Xπ β π₯ if(π = π, (-β(,)π¦), β))) | |
6 | 1, 2, 3, 4, 5 | hspdifhsp 45010 | . 2 β’ (π β Xπ β π ((π΄βπ)[,)(π΅βπ)) = β© π β π ((π(π»βπ)(π΅βπ)) β (π(π»βπ)(π΄βπ)))) |
7 | 1 | vonmea 44968 | . . . 4 β’ (π β (volnβπ) β Meas) |
8 | hoimbllem.s | . . . 4 β’ π = dom (volnβπ) | |
9 | 7, 8 | dmmeasal 44846 | . . 3 β’ (π β π β SAlg) |
10 | fict 9613 | . . . 4 β’ (π β Fin β π βΌ Ο) | |
11 | 1, 10 | syl 17 | . . 3 β’ (π β π βΌ Ο) |
12 | 9 | adantr 481 | . . . 4 β’ ((π β§ π β π) β π β SAlg) |
13 | 1 | adantr 481 | . . . . . 6 β’ ((π β§ π β π) β π β Fin) |
14 | simpr 485 | . . . . . 6 β’ ((π β§ π β π) β π β π) | |
15 | 4 | adantr 481 | . . . . . . 7 β’ ((π β§ π β π) β π΅:πβΆβ) |
16 | 15, 14 | ffvelcdmd 7056 | . . . . . 6 β’ ((π β§ π β π) β (π΅βπ) β β) |
17 | 5, 13, 14, 16 | hspmbl 45023 | . . . . 5 β’ ((π β§ π β π) β (π(π»βπ)(π΅βπ)) β dom (volnβπ)) |
18 | 8 | eqcomi 2740 | . . . . . 6 β’ dom (volnβπ) = π |
19 | 18 | a1i 11 | . . . . 5 β’ ((π β§ π β π) β dom (volnβπ) = π) |
20 | 17, 19 | eleqtrd 2834 | . . . 4 β’ ((π β§ π β π) β (π(π»βπ)(π΅βπ)) β π) |
21 | 3 | ffvelcdmda 7055 | . . . . . 6 β’ ((π β§ π β π) β (π΄βπ) β β) |
22 | 5, 13, 14, 21 | hspmbl 45023 | . . . . 5 β’ ((π β§ π β π) β (π(π»βπ)(π΄βπ)) β dom (volnβπ)) |
23 | 22, 19 | eleqtrd 2834 | . . . 4 β’ ((π β§ π β π) β (π(π»βπ)(π΄βπ)) β π) |
24 | saldifcl2 44722 | . . . 4 β’ ((π β SAlg β§ (π(π»βπ)(π΅βπ)) β π β§ (π(π»βπ)(π΄βπ)) β π) β ((π(π»βπ)(π΅βπ)) β (π(π»βπ)(π΄βπ))) β π) | |
25 | 12, 20, 23, 24 | syl3anc 1371 | . . 3 β’ ((π β§ π β π) β ((π(π»βπ)(π΅βπ)) β (π(π»βπ)(π΄βπ))) β π) |
26 | 9, 11, 2, 25 | saliincl 44721 | . 2 β’ (π β β© π β π ((π(π»βπ)(π΅βπ)) β (π(π»βπ)(π΄βπ))) β π) |
27 | 6, 26 | eqeltrd 2832 | 1 β’ (π β Xπ β π ((π΄βπ)[,)(π΅βπ)) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2939 β cdif 3925 β c0 4302 ifcif 4506 β© ciin 4975 class class class wbr 5125 β¦ cmpt 5208 dom cdm 5653 βΆwf 6512 βcfv 6516 (class class class)co 7377 β cmpo 7379 Οcom 7822 Xcixp 8857 βΌ cdom 8903 Fincfn 8905 βcr 11074 -βcmnf 11211 (,)cioo 13289 [,)cico 13291 SAlgcsalg 44702 volncvoln 44932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cc 10395 ax-ac2 10423 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-disj 5091 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-tpos 8177 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-omul 8437 df-er 8670 df-map 8789 df-pm 8790 df-ixp 8858 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9470 df-dju 9861 df-card 9899 df-acn 9902 df-ac 10076 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-q 12898 df-rp 12940 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13450 df-fzo 13593 df-fl 13722 df-seq 13932 df-exp 13993 df-hash 14256 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-clim 15397 df-rlim 15398 df-sum 15598 df-prod 15815 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-starv 17177 df-tset 17181 df-ple 17182 df-ds 17184 df-unif 17185 df-rest 17333 df-0g 17352 df-topgen 17354 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-grp 18780 df-minusg 18781 df-subg 18954 df-cmn 19593 df-abl 19594 df-mgp 19926 df-ur 19943 df-ring 19995 df-cring 19996 df-oppr 20078 df-dvdsr 20099 df-unit 20100 df-invr 20130 df-dvr 20141 df-drng 20242 df-psmet 20840 df-xmet 20841 df-met 20842 df-bl 20843 df-mopn 20844 df-cnfld 20849 df-top 22295 df-topon 22312 df-bases 22348 df-cmp 22790 df-ovol 24880 df-vol 24881 df-salg 44703 df-sumge0 44757 df-mea 44844 df-ome 44884 df-caragen 44886 df-ovoln 44931 df-voln 44933 |
This theorem is referenced by: hoimbl 45025 |
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