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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocat | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.) |
Ref | Expression |
---|---|
lhpocat.o | β’ β₯ = (ocβπΎ) |
lhpocat.a | β’ π΄ = (AtomsβπΎ) |
lhpocat.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpocat | β’ ((πΎ β HL β§ π β π») β ( β₯ βπ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . 2 β’ ((πΎ β HL β§ π β π») β π β π») | |
2 | eqid 2736 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | lhpocat.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | 2, 3 | lhpbase 38317 | . . 3 β’ (π β π» β π β (BaseβπΎ)) |
5 | lhpocat.o | . . . 4 β’ β₯ = (ocβπΎ) | |
6 | lhpocat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | 2, 5, 6, 3 | lhpoc 38333 | . . 3 β’ ((πΎ β HL β§ π β (BaseβπΎ)) β (π β π» β ( β₯ βπ) β π΄)) |
8 | 4, 7 | sylan2 593 | . 2 β’ ((πΎ β HL β§ π β π») β (π β π» β ( β₯ βπ) β π΄)) |
9 | 1, 8 | mpbid 231 | 1 β’ ((πΎ β HL 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 396 = wceq 1540 β wcel 2105 βcfv 6480 Basecbs 17010 occoc 17068 Atomscatm 37581 HLchlt 37668 LHypclh 38303 |
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 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 |
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-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 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-proset 18111 df-poset 18129 df-plt 18146 df-lub 18162 df-glb 18163 df-p0 18241 df-p1 18242 df-oposet 37494 df-ol 37496 df-oml 37497 df-covers 37584 df-ats 37585 df-hlat 37669 df-lhyp 38307 |
This theorem is referenced by: lhpocnel 38337 lhpmod2i2 38357 lhpmod6i1 38358 dihat 39654 |
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