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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 2726 | . . 3 β’ (ODualβπΎ) = (ODualβπΎ) | |
2 | posglbdg.l | . . 3 β’ β€ = (leβπΎ) | |
3 | 1, 2 | oduleval 18254 | . 2 β’ β‘ β€ = (leβ(ODualβπΎ)) |
4 | posglbdg.b | . . 3 β’ (π β π΅ = (BaseβπΎ)) | |
5 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
6 | 1, 5 | odubas 18256 | . . 3 β’ (BaseβπΎ) = (Baseβ(ODualβπΎ)) |
7 | 4, 6 | eqtrdi 2782 | . 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 2726 | . . . . 5 β’ (glbβπΎ) = (glbβπΎ) | |
11 | 1, 10 | odulub 18372 | . . . 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 2766 | . 2 β’ (π β πΊ = (lubβ(ODualβπΎ))) |
14 | 1 | odupos 18293 | . . 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 3472 | . . . . 5 β’ π₯ β V | |
20 | brcnvg 5873 | . . . . 5 β’ ((π₯ β V β§ π β π΅) β (π₯β‘ β€ π β π β€ π₯)) | |
21 | 19, 17, 20 | sylancr 586 | . . . 4 β’ (π β (π₯β‘ β€ π β π β€ π₯)) |
22 | 21 | adantr 480 | . . 3 β’ ((π β§ π₯ β π) β (π₯β‘ β€ π β π β€ π₯)) |
23 | 18, 22 | mpbird 257 | . 2 β’ ((π β§ π₯ β π) β π₯β‘ β€ π) |
24 | vex 3472 | . . . . . 6 β’ π¦ β V | |
25 | 19, 24 | brcnv 5876 | . . . . 5 β’ (π₯β‘ β€ π¦ β π¦ β€ π₯) |
26 | 25 | ralbii 3087 | . . . 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 5873 | . . . . 5 β’ ((π β π΅ β§ π¦ β V) β (πβ‘ β€ π¦ β π¦ β€ π)) | |
30 | 17, 24, 29 | sylancl 585 | . . . 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 257 | . 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 18379 | 1 β’ (π β (πΊβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 Vcvv 3468 β wss 3943 class class class wbr 5141 β‘ccnv 5668 βcfv 6537 Basecbs 17153 lecple 17213 ODualcodu 18251 Posetcpo 18272 lubclub 18274 glbcglb 18275 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-dec 12682 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ple 17226 df-odu 18252 df-proset 18260 df-poset 18278 df-lub 18311 df-glb 18312 |
This theorem is referenced by: mrelatglb 18525 mrelatglb0 18526 glbsscl 47865 ipoglb 47887 |
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