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Mathbox for Zhi Wang |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > topclat | Structured version Visualization version GIF version |
Description: A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
topclat.i | β’ πΌ = (toIncβπ½) |
Ref | Expression |
---|---|
topclat | β’ (π½ β Top β πΌ β CLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topclat.i | . . 3 β’ πΌ = (toIncβπ½) | |
2 | 1 | ipobas 18493 | . 2 β’ (π½ β Top β π½ = (BaseβπΌ)) |
3 | eqidd 2727 | . 2 β’ (π½ β Top β (lubβπΌ) = (lubβπΌ)) | |
4 | eqidd 2727 | . 2 β’ (π½ β Top β (glbβπΌ) = (glbβπΌ)) | |
5 | 1 | ipopos 18498 | . . 3 β’ πΌ β Poset |
6 | 5 | a1i 11 | . 2 β’ (π½ β Top β πΌ β Poset) |
7 | uniopn 22749 | . . 3 β’ ((π½ β Top β§ π₯ β π½) β βͺ π₯ β π½) | |
8 | simpl 482 | . . . 4 β’ ((π½ β Top β§ π₯ β π½) β π½ β Top) | |
9 | simpr 484 | . . . 4 β’ ((π½ β Top β§ π₯ β π½) β π₯ β π½) | |
10 | eqidd 2727 | . . . 4 β’ ((π½ β Top β§ π₯ β π½) β (lubβπΌ) = (lubβπΌ)) | |
11 | intmin 4965 | . . . . . 6 β’ (βͺ π₯ β π½ β β© {π¦ β π½ β£ βͺ π₯ β π¦} = βͺ π₯) | |
12 | 11 | eqcomd 2732 | . . . . 5 β’ (βͺ π₯ β π½ β βͺ π₯ = β© {π¦ β π½ β£ βͺ π₯ β π¦}) |
13 | 7, 12 | syl 17 | . . . 4 β’ ((π½ β Top β§ π₯ β π½) β βͺ π₯ = β© {π¦ β π½ β£ βͺ π₯ β π¦}) |
14 | 1, 8, 9, 10, 13 | ipolubdm 47868 | . . 3 β’ ((π½ β Top β§ π₯ β π½) β (π₯ β dom (lubβπΌ) β βͺ π₯ β π½)) |
15 | 7, 14 | mpbird 257 | . 2 β’ ((π½ β Top β§ π₯ β π½) β π₯ β dom (lubβπΌ)) |
16 | ssrab2 4072 | . . . 4 β’ {π¦ β π½ β£ π¦ β β© π₯} β π½ | |
17 | uniopn 22749 | . . . 4 β’ ((π½ β Top β§ {π¦ β π½ β£ π¦ β β© π₯} β π½) β βͺ {π¦ β π½ β£ π¦ β β© π₯} β π½) | |
18 | 8, 16, 17 | sylancl 585 | . . 3 β’ ((π½ β Top β§ π₯ β π½) β βͺ {π¦ β π½ β£ π¦ β β© π₯} β π½) |
19 | eqidd 2727 | . . . 4 β’ ((π½ β Top β§ π₯ β π½) β (glbβπΌ) = (glbβπΌ)) | |
20 | eqidd 2727 | . . . 4 β’ ((π½ β Top β§ π₯ β π½) β βͺ {π¦ β π½ β£ π¦ β β© π₯} = βͺ {π¦ β π½ β£ π¦ β β© π₯}) | |
21 | 1, 8, 9, 19, 20 | ipoglbdm 47871 | . . 3 β’ ((π½ β Top β§ π₯ β π½) β (π₯ β dom (glbβπΌ) β βͺ {π¦ β π½ β£ π¦ β β© π₯} β π½)) |
22 | 18, 21 | mpbird 257 | . 2 β’ ((π½ β Top β§ π₯ β π½) β π₯ β dom (glbβπΌ)) |
23 | 2, 3, 4, 6, 15, 22 | isclatd 47864 | 1 β’ (π½ β Top β πΌ β CLat) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3426 β wss 3943 βͺ cuni 4902 β© cint 4943 dom cdm 5669 βcfv 6536 Posetcpo 18269 lubclub 18271 glbcglb 18272 CLatccla 18460 toInccipo 18489 Topctop 22745 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-tset 17222 df-ple 17223 df-ocomp 17224 df-proset 18257 df-poset 18275 df-lub 18308 df-glb 18309 df-clat 18461 df-ipo 18490 df-top 22746 |
This theorem is referenced by: topdlat 47885 |
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