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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 2724 | . . 3 β’ (1.βπΎ) = (1.βπΎ) | |
3 | eqid 2724 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
4 | lhpoc.h | . . 3 β’ π» = (LHypβπΎ) | |
5 | 1, 2, 3, 4 | islhp2 39371 | . 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 38846 | . 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 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 Basecbs 17149 occoc 17210 1.cp1 18385 β ccvr 38635 Atomscatm 38636 HLchlt 38723 LHypclh 39358 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-p0 18386 df-p1 18387 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-hlat 38724 df-lhyp 39362 |
This theorem is referenced by: lhpoc2N 39389 lhpocnle 39390 lhpocat 39391 |
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