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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpoc | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.) |
Ref | Expression |
---|---|
lhpoc.b | β’ π΅ = (BaseβπΎ) |
lhpoc.o | β’ β₯ = (ocβπΎ) |
lhpoc.a | β’ π΄ = (AtomsβπΎ) |
lhpoc.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpoc | β’ ((πΎ β HL β§ π β π΅) β (π β π» β ( β₯ βπ) β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpoc.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2728 | . . 3 β’ (1.βπΎ) = (1.βπΎ) | |
3 | eqid 2728 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
4 | lhpoc.h | . . 3 β’ π» = (LHypβπΎ) | |
5 | 1, 2, 3, 4 | islhp2 39470 | . 2 β’ ((πΎ β HL β§ π β π΅) β (π β π» β π( β βπΎ)(1.βπΎ))) |
6 | lhpoc.o | . . 3 β’ β₯ = (ocβπΎ) | |
7 | lhpoc.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
8 | 1, 2, 6, 3, 7 | 1cvrco 38945 | . 2 β’ ((πΎ β HL β§ π β π΅) β (π( β βπΎ)(1.βπΎ) β ( β₯ βπ) β π΄)) |
9 | 5, 8 | bitrd 279 | 1 β’ ((πΎ β HL β§ π β π΅) β (π β π» β ( β₯ βπ) β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 Basecbs 17179 occoc 17240 1.cp1 18415 β ccvr 38734 Atomscatm 38735 HLchlt 38822 LHypclh 39457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-p0 18416 df-p1 18417 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-hlat 38823 df-lhyp 39461 |
This theorem is referenced by: lhpoc2N 39488 lhpocnle 39489 lhpocat 39490 |
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