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Mirrors > Home > MPE Home > Th. List > icopnfcld | Structured version Visualization version GIF version |
Description: Right-unbounded closed intervals are closed sets of the standard topology on β. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
icopnfcld | β’ (π΄ β β β (π΄[,)+β) β (Clsdβ(topGenβran (,)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 11301 | . . . . . 6 β’ -β β β* | |
2 | 1 | a1i 11 | . . . . 5 β’ (π΄ β β β -β β β*) |
3 | rexr 11290 | . . . . 5 β’ (π΄ β β β π΄ β β*) | |
4 | pnfxr 11298 | . . . . . 6 β’ +β β β* | |
5 | 4 | a1i 11 | . . . . 5 β’ (π΄ β β β +β β β*) |
6 | mnflt 13135 | . . . . 5 β’ (π΄ β β β -β < π΄) | |
7 | ltpnf 13132 | . . . . 5 β’ (π΄ β β β π΄ < +β) | |
8 | df-ioo 13360 | . . . . . 6 β’ (,) = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯ < π§ β§ π§ < π¦)}) | |
9 | df-ico 13362 | . . . . . 6 β’ [,) = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯ β€ π§ β§ π§ < π¦)}) | |
10 | xrlenlt 11309 | . . . . . 6 β’ ((π΄ β β* β§ π€ β β*) β (π΄ β€ π€ β Β¬ π€ < π΄)) | |
11 | xrlttr 13151 | . . . . . 6 β’ ((π€ β β* β§ π΄ β β* β§ +β β β*) β ((π€ < π΄ β§ π΄ < +β) β π€ < +β)) | |
12 | xrltletr 13168 | . . . . . 6 β’ ((-β β β* β§ π΄ β β* β§ π€ β β*) β ((-β < π΄ β§ π΄ β€ π€) β -β < π€)) | |
13 | 8, 9, 10, 8, 11, 12 | ixxun 13372 | . . . . 5 β’ (((-β β β* β§ π΄ β β* β§ +β β β*) β§ (-β < π΄ β§ π΄ < +β)) β ((-β(,)π΄) βͺ (π΄[,)+β)) = (-β(,)+β)) |
14 | 2, 3, 5, 6, 7, 13 | syl32anc 1375 | . . . 4 β’ (π΄ β β β ((-β(,)π΄) βͺ (π΄[,)+β)) = (-β(,)+β)) |
15 | ioomax 13431 | . . . 4 β’ (-β(,)+β) = β | |
16 | 14, 15 | eqtrdi 2781 | . . 3 β’ (π΄ β β β ((-β(,)π΄) βͺ (π΄[,)+β)) = β) |
17 | ioossre 13417 | . . . 4 β’ (-β(,)π΄) β β | |
18 | 8, 9, 10 | ixxdisj 13371 | . . . . 5 β’ ((-β β β* β§ π΄ β β* β§ +β β β*) β ((-β(,)π΄) β© (π΄[,)+β)) = β ) |
19 | 1, 3, 5, 18 | mp3an2i 1462 | . . . 4 β’ (π΄ β β β ((-β(,)π΄) β© (π΄[,)+β)) = β ) |
20 | uneqdifeq 4488 | . . . 4 β’ (((-β(,)π΄) β β β§ ((-β(,)π΄) β© (π΄[,)+β)) = β ) β (((-β(,)π΄) βͺ (π΄[,)+β)) = β β (β β (-β(,)π΄)) = (π΄[,)+β))) | |
21 | 17, 19, 20 | sylancr 585 | . . 3 β’ (π΄ β β β (((-β(,)π΄) βͺ (π΄[,)+β)) = β β (β β (-β(,)π΄)) = (π΄[,)+β))) |
22 | 16, 21 | mpbid 231 | . 2 β’ (π΄ β β β (β β (-β(,)π΄)) = (π΄[,)+β)) |
23 | retop 24696 | . . 3 β’ (topGenβran (,)) β Top | |
24 | iooretop 24700 | . . 3 β’ (-β(,)π΄) β (topGenβran (,)) | |
25 | uniretop 24697 | . . . 4 β’ β = βͺ (topGenβran (,)) | |
26 | 25 | opncld 22955 | . . 3 β’ (((topGenβran (,)) β Top β§ (-β(,)π΄) β (topGenβran (,))) β (β β (-β(,)π΄)) β (Clsdβ(topGenβran (,)))) |
27 | 23, 24, 26 | mp2an 690 | . 2 β’ (β β (-β(,)π΄)) β (Clsdβ(topGenβran (,))) |
28 | 22, 27 | eqeltrrdi 2834 | 1 β’ (π΄ β β β (π΄[,)+β) β (Clsdβ(topGenβran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β cdif 3936 βͺ cun 3937 β© cin 3938 β wss 3939 β c0 4318 class class class wbr 5143 ran crn 5673 βcfv 6543 (class class class)co 7416 βcr 11137 +βcpnf 11275 -βcmnf 11276 β*cxr 11277 < clt 11278 β€ cle 11279 (,)cioo 13356 [,)cico 13358 topGenctg 17418 Topctop 22813 Clsdccld 22938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-ioo 13360 df-ico 13362 df-topgen 17424 df-top 22814 df-bases 22867 df-cld 22941 |
This theorem is referenced by: sxbrsigalem3 33949 orvcgteel 34144 dvasin 37234 dvacos 37235 dvreasin 37236 dvreacos 37237 rfcnpre3 44460 |
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