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Mirrors > Home > MPE Home > Th. List > poslubdg | Structured version Visualization version GIF version |
Description: Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
poslubdg.l | β’ β€ = (leβπΎ) |
poslubdg.b | β’ (π β π΅ = (BaseβπΎ)) |
poslubdg.u | β’ (π β π = (lubβπΎ)) |
poslubdg.k | β’ (π β πΎ β Poset) |
poslubdg.s | β’ (π β π β π΅) |
poslubdg.t | β’ (π β π β π΅) |
poslubdg.ub | β’ ((π β§ π₯ β π) β π₯ β€ π) |
poslubdg.le | β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π₯ β€ π¦) β π β€ π¦) |
Ref | Expression |
---|---|
poslubdg | β’ (π β (πβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poslubdg.u | . . 3 β’ (π β π = (lubβπΎ)) | |
2 | 1 | fveq1d 6827 | . 2 β’ (π β (πβπ) = ((lubβπΎ)βπ)) |
3 | poslubdg.l | . . 3 β’ β€ = (leβπΎ) | |
4 | eqid 2736 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | eqid 2736 | . . 3 β’ (lubβπΎ) = (lubβπΎ) | |
6 | poslubdg.k | . . 3 β’ (π β πΎ β Poset) | |
7 | poslubdg.s | . . . 4 β’ (π β π β π΅) | |
8 | poslubdg.b | . . . 4 β’ (π β π΅ = (BaseβπΎ)) | |
9 | 7, 8 | sseqtrd 3972 | . . 3 β’ (π β π β (BaseβπΎ)) |
10 | poslubdg.t | . . . 4 β’ (π β π β π΅) | |
11 | 10, 8 | eleqtrd 2839 | . . 3 β’ (π β π β (BaseβπΎ)) |
12 | poslubdg.ub | . . 3 β’ ((π β§ π₯ β π) β π₯ β€ π) | |
13 | 8 | eleq2d 2822 | . . . . . 6 β’ (π β (π¦ β π΅ β π¦ β (BaseβπΎ))) |
14 | 13 | biimpar 478 | . . . . 5 β’ ((π β§ π¦ β (BaseβπΎ)) β π¦ β π΅) |
15 | 14 | 3adant3 1131 | . . . 4 β’ ((π β§ π¦ β (BaseβπΎ) β§ βπ₯ β π π₯ β€ π¦) β π¦ β π΅) |
16 | poslubdg.le | . . . 4 β’ ((π β§ π¦ β π΅ β§ βπ₯ β π π₯ β€ π¦) β π β€ π¦) | |
17 | 15, 16 | syld3an2 1410 | . . 3 β’ ((π β§ π¦ β (BaseβπΎ) β§ βπ₯ β π π₯ β€ π¦) β π β€ π¦) |
18 | 3, 4, 5, 6, 9, 11, 12, 17 | poslubd 18228 | . 2 β’ (π β ((lubβπΎ)βπ) = π) |
19 | 2, 18 | eqtrd 2776 | 1 β’ (π β (πβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3061 β wss 3898 class class class wbr 5092 βcfv 6479 Basecbs 17009 lecple 17066 Posetcpo 18122 lubclub 18124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-proset 18110 df-poset 18128 df-lub 18161 |
This theorem is referenced by: posglbdg 18230 mrelatlub 18377 ipolub 46625 |
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