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Mirrors > Home > MPE Home > Th. List > Mathboxes > toplatlub | Structured version Visualization version GIF version |
Description: Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
topclat.i | β’ πΌ = (toIncβπ½) |
toplatlub.j | β’ (π β π½ β Top) |
toplatlub.s | β’ (π β π β π½) |
toplatlub.u | β’ π = (lubβπΌ) |
Ref | Expression |
---|---|
toplatlub | β’ (π β (πβπ) = βͺ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topclat.i | . 2 β’ πΌ = (toIncβπ½) | |
2 | toplatlub.j | . 2 β’ (π β π½ β Top) | |
3 | toplatlub.s | . 2 β’ (π β π β π½) | |
4 | toplatlub.u | . . 3 β’ π = (lubβπΌ) | |
5 | 4 | a1i 11 | . 2 β’ (π β π = (lubβπΌ)) |
6 | uniopn 22817 | . . . 4 β’ ((π½ β Top β§ π β π½) β βͺ π β π½) | |
7 | 2, 3, 6 | syl2anc 582 | . . 3 β’ (π β βͺ π β π½) |
8 | intmin 4966 | . . . 4 β’ (βͺ π β π½ β β© {π₯ β π½ β£ βͺ π β π₯} = βͺ π) | |
9 | 8 | eqcomd 2731 | . . 3 β’ (βͺ π β π½ β βͺ π = β© {π₯ β π½ β£ βͺ π β π₯}) |
10 | 7, 9 | syl 17 | . 2 β’ (π β βͺ π = β© {π₯ β π½ β£ βͺ π β π₯}) |
11 | 1, 2, 3, 5, 10, 7 | ipolub 48111 | 1 β’ (π β (πβπ) = βͺ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3419 β wss 3939 βͺ cuni 4903 β© cint 4944 βcfv 6543 lubclub 18300 toInccipo 18518 Topctop 22813 |
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-rep 5280 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 |
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-int 4945 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-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-tset 17251 df-ple 17252 df-ocomp 17253 df-proset 18286 df-poset 18304 df-lub 18337 df-ipo 18519 df-top 22814 |
This theorem is referenced by: (None) |
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