Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iocborel | Structured version Visualization version GIF version |
Description: A left-open, right-closed interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
iocborel.a | β’ (π β π΄ β β*) |
iocborel.c | β’ (π β πΆ β β) |
iocborel.t | β’ π½ = (topGenβran (,)) |
iocborel.b | β’ π΅ = (SalGenβπ½) |
Ref | Expression |
---|---|
iocborel | β’ (π β (π΄(,]πΆ) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iocborel.a | . . . 4 β’ (π β π΄ β β*) | |
2 | iocborel.c | . . . 4 β’ (π β πΆ β β) | |
3 | 1, 2 | iooiinioc 43438 | . . 3 β’ (π β β© π β β (π΄(,)(πΆ + (1 / π))) = (π΄(,]πΆ)) |
4 | 3 | eqcomd 2742 | . 2 β’ (π β (π΄(,]πΆ) = β© π β β (π΄(,)(πΆ + (1 / π)))) |
5 | iocborel.t | . . . . . . 7 β’ π½ = (topGenβran (,)) | |
6 | iocborel.b | . . . . . . 7 β’ π΅ = (SalGenβπ½) | |
7 | 5, 6 | bor1sal 44238 | . . . . . 6 β’ π΅ β SAlg |
8 | 7 | a1i 11 | . . . . 5 β’ (β€ β π΅ β SAlg) |
9 | nnct 13802 | . . . . . 6 β’ β βΌ Ο | |
10 | 9 | a1i 11 | . . . . 5 β’ (β€ β β βΌ Ο) |
11 | nnn0 43260 | . . . . . 6 β’ β β β | |
12 | 11 | a1i 11 | . . . . 5 β’ (β€ β β β β ) |
13 | 5, 6 | iooborel 44234 | . . . . . 6 β’ (π΄(,)(πΆ + (1 / π))) β π΅ |
14 | 13 | a1i 11 | . . . . 5 β’ ((β€ β§ π β β) β (π΄(,)(πΆ + (1 / π))) β π΅) |
15 | 8, 10, 12, 14 | saliincl 44210 | . . . 4 β’ (β€ β β© π β β (π΄(,)(πΆ + (1 / π))) β π΅) |
16 | 15 | mptru 1547 | . . 3 β’ β© π β β (π΄(,)(πΆ + (1 / π))) β π΅ |
17 | 16 | a1i 11 | . 2 β’ (π β β© π β β (π΄(,)(πΆ + (1 / π))) β π΅) |
18 | 4, 17 | eqeltrd 2837 | 1 β’ (π β (π΄(,]πΆ) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β€wtru 1541 β wcel 2105 β wne 2940 β c0 4269 β© ciin 4942 class class class wbr 5092 ran crn 5621 βcfv 6479 (class class class)co 7337 Οcom 7780 βΌ cdom 8802 βcr 10971 1c1 10973 + caddc 10975 β*cxr 11109 / cdiv 11733 βcn 12074 (,)cioo 13180 (,]cioc 13181 topGenctg 17245 SAlgcsalg 44193 SalGencsalgen 44197 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-inf2 9498 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-card 9796 df-acn 9799 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-q 12790 df-rp 12832 df-ioo 13184 df-ioc 13185 df-fl 13613 df-topgen 17251 df-top 22149 df-bases 22202 df-salg 44194 df-salgen 44198 |
This theorem is referenced by: incsmflem 44624 decsmflem 44649 smfsuplem2 44695 |
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