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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 2728 | . . 3 β’ (ODualβπΎ) = (ODualβπΎ) | |
2 | posglbdg.l | . . 3 β’ β€ = (leβπΎ) | |
3 | 1, 2 | oduleval 18288 | . 2 β’ β‘ β€ = (leβ(ODualβπΎ)) |
4 | posglbdg.b | . . 3 β’ (π β π΅ = (BaseβπΎ)) | |
5 | eqid 2728 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
6 | 1, 5 | odubas 18290 | . . 3 β’ (BaseβπΎ) = (Baseβ(ODualβπΎ)) |
7 | 4, 6 | eqtrdi 2784 | . 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 2728 | . . . . 5 β’ (glbβπΎ) = (glbβπΎ) | |
11 | 1, 10 | odulub 18406 | . . . 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 2768 | . 2 β’ (π β πΊ = (lubβ(ODualβπΎ))) |
14 | 1 | odupos 18327 | . . 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 3477 | . . . . 5 β’ π₯ β V | |
20 | brcnvg 5886 | . . . . 5 β’ ((π₯ β V β§ π β π΅) β (π₯β‘ β€ π β π β€ π₯)) | |
21 | 19, 17, 20 | sylancr 585 | . . . 4 β’ (π β (π₯β‘ β€ π β π β€ π₯)) |
22 | 21 | adantr 479 | . . 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 3477 | . . . . . 6 β’ π¦ β V | |
25 | 19, 24 | brcnv 5889 | . . . . 5 β’ (π₯β‘ β€ π¦ β π¦ β€ π₯) |
26 | 25 | ralbii 3090 | . . . 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 1402 | . . 3 β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π₯β‘ β€ π¦) β π¦ β€ π) |
29 | brcnvg 5886 | . . . . 5 β’ ((π β π΅ β§ π¦ β V) β (πβ‘ β€ π¦ β π¦ β€ π)) | |
30 | 17, 24, 29 | sylancl 584 | . . . 4 β’ (π β (πβ‘ β€ π¦ β π¦ β€ π)) |
31 | 30 | 3ad2ant1 1130 | . . 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 18413 | 1 β’ (π β (πΊβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 Vcvv 3473 β wss 3949 class class class wbr 5152 β‘ccnv 5681 βcfv 6553 Basecbs 17187 lecple 17247 ODualcodu 18285 Posetcpo 18306 lubclub 18308 glbcglb 18309 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-dec 12716 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ple 17260 df-odu 18286 df-proset 18294 df-poset 18312 df-lub 18345 df-glb 18346 |
This theorem is referenced by: mrelatglb 18559 mrelatglb0 18560 glbsscl 48058 ipoglb 48080 |
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