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Mirrors > Home > MPE Home > Th. List > posglbdg | Structured version Visualization version GIF version |
Description: Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
posglbdg.l | β’ β€ = (leβπΎ) |
posglbdg.b | β’ (π β π΅ = (BaseβπΎ)) |
posglbdg.g | β’ (π β πΊ = (glbβπΎ)) |
posglbdg.k | β’ (π β πΎ β Poset) |
posglbdg.s | β’ (π β π β π΅) |
posglbdg.t | β’ (π β π β π΅) |
posglbdg.lb | β’ ((π β§ π₯ β π) β π β€ π₯) |
posglbdg.gt | β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π¦ β€ π₯) β π¦ β€ π) |
Ref | Expression |
---|---|
posglbdg | β’ (π β (πΊβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 β’ (ODualβπΎ) = (ODualβπΎ) | |
2 | posglbdg.l | . . 3 β’ β€ = (leβπΎ) | |
3 | 1, 2 | oduleval 18238 | . 2 β’ β‘ β€ = (leβ(ODualβπΎ)) |
4 | posglbdg.b | . . 3 β’ (π β π΅ = (BaseβπΎ)) | |
5 | eqid 2732 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
6 | 1, 5 | odubas 18240 | . . 3 β’ (BaseβπΎ) = (Baseβ(ODualβπΎ)) |
7 | 4, 6 | eqtrdi 2788 | . 2 β’ (π β π΅ = (Baseβ(ODualβπΎ))) |
8 | posglbdg.g | . . 3 β’ (π β πΊ = (glbβπΎ)) | |
9 | posglbdg.k | . . . 4 β’ (π β πΎ β Poset) | |
10 | eqid 2732 | . . . . 5 β’ (glbβπΎ) = (glbβπΎ) | |
11 | 1, 10 | odulub 18356 | . . . 4 β’ (πΎ β Poset β (glbβπΎ) = (lubβ(ODualβπΎ))) |
12 | 9, 11 | syl 17 | . . 3 β’ (π β (glbβπΎ) = (lubβ(ODualβπΎ))) |
13 | 8, 12 | eqtrd 2772 | . 2 β’ (π β πΊ = (lubβ(ODualβπΎ))) |
14 | 1 | odupos 18277 | . . 3 β’ (πΎ β Poset β (ODualβπΎ) β Poset) |
15 | 9, 14 | syl 17 | . 2 β’ (π β (ODualβπΎ) β Poset) |
16 | posglbdg.s | . 2 β’ (π β π β π΅) | |
17 | posglbdg.t | . 2 β’ (π β π β π΅) | |
18 | posglbdg.lb | . . 3 β’ ((π β§ π₯ β π) β π β€ π₯) | |
19 | vex 3478 | . . . . 5 β’ π₯ β V | |
20 | brcnvg 5877 | . . . . 5 β’ ((π₯ β V β§ π β π΅) β (π₯β‘ β€ π β π β€ π₯)) | |
21 | 19, 17, 20 | sylancr 587 | . . . 4 β’ (π β (π₯β‘ β€ π β π β€ π₯)) |
22 | 21 | adantr 481 | . . 3 β’ ((π β§ π₯ β π) β (π₯β‘ β€ π β π β€ π₯)) |
23 | 18, 22 | mpbird 256 | . 2 β’ ((π β§ π₯ β π) β π₯β‘ β€ π) |
24 | vex 3478 | . . . . . 6 β’ π¦ β V | |
25 | 19, 24 | brcnv 5880 | . . . . 5 β’ (π₯β‘ β€ π¦ β π¦ β€ π₯) |
26 | 25 | ralbii 3093 | . . . 4 β’ (βπ₯ β π π₯β‘ β€ π¦ β βπ₯ β π π¦ β€ π₯) |
27 | posglbdg.gt | . . . 4 β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π¦ β€ π₯) β π¦ β€ π) | |
28 | 26, 27 | syl3an3b 1405 | . . 3 β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π₯β‘ β€ π¦) β π¦ β€ π) |
29 | brcnvg 5877 | . . . . 5 β’ ((π β π΅ β§ π¦ β V) β (πβ‘ β€ π¦ β π¦ β€ π)) | |
30 | 17, 24, 29 | sylancl 586 | . . . 4 β’ (π β (πβ‘ β€ π¦ β π¦ β€ π)) |
31 | 30 | 3ad2ant1 1133 | . . 3 β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π₯β‘ β€ π¦) β (πβ‘ β€ π¦ β π¦ β€ π)) |
32 | 28, 31 | mpbird 256 | . 2 β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π₯β‘ β€ π¦) β πβ‘ β€ π¦) |
33 | 3, 7, 13, 15, 16, 17, 23, 32 | poslubdg 18363 | 1 β’ (π β (πΊβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β wss 3947 class class class wbr 5147 β‘ccnv 5674 βcfv 6540 Basecbs 17140 lecple 17200 ODualcodu 18235 Posetcpo 18256 lubclub 18258 glbcglb 18259 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-dec 12674 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ple 17213 df-odu 18236 df-proset 18244 df-poset 18262 df-lub 18295 df-glb 18296 |
This theorem is referenced by: mrelatglb 18509 mrelatglb0 18510 glbsscl 47547 ipoglb 47569 |
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