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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 6904 | . 2 β’ (π β (πβπ) = ((lubβπΎ)βπ)) |
3 | poslubdg.l | . . 3 β’ β€ = (leβπΎ) | |
4 | eqid 2728 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | eqid 2728 | . . 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 4022 | . . 3 β’ (π β π β (BaseβπΎ)) |
10 | poslubdg.t | . . . 4 β’ (π β π β π΅) | |
11 | 10, 8 | eleqtrd 2831 | . . 3 β’ (π β π β (BaseβπΎ)) |
12 | poslubdg.ub | . . 3 β’ ((π β§ π₯ β π) β π₯ β€ π) | |
13 | 8 | eleq2d 2815 | . . . . . 6 β’ (π β (π¦ β π΅ β π¦ β (BaseβπΎ))) |
14 | 13 | biimpar 476 | . . . . 5 β’ ((π β§ π¦ β (BaseβπΎ)) β π¦ β π΅) |
15 | 14 | 3adant3 1129 | . . . 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 1408 | . . 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 18412 | . 2 β’ (π β ((lubβπΎ)βπ) = π) |
19 | 2, 18 | eqtrd 2768 | 1 β’ (π β (πβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 β wss 3949 class class class wbr 5152 βcfv 6553 Basecbs 17187 lecple 17247 Posetcpo 18306 lubclub 18308 |
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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-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-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-id 5580 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-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-proset 18294 df-poset 18312 df-lub 18345 |
This theorem is referenced by: posglbdg 18414 mrelatlub 18561 ipolub 48077 |
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