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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlsupr | Structured version Visualization version GIF version |
Description: A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.) |
Ref | Expression |
---|---|
hlsupr.l | β’ β€ = (leβπΎ) |
hlsupr.j | β’ β¨ = (joinβπΎ) |
hlsupr.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlsupr | β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β π) β βπ β π΄ (π β π β§ π β π β§ π β€ (π β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | hlsupr.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | hlsupr.j | . . . 4 β’ β¨ = (joinβπΎ) | |
4 | hlsupr.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | hlsuprexch 37873 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β ((π β π β βπ β π΄ (π β π β§ π β π β§ π β€ (π β¨ π))) β§ βπ β (BaseβπΎ)((Β¬ π β€ π β§ π β€ (π β¨ π)) β π β€ (π β¨ π)))) |
6 | 5 | simpld 496 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β π β βπ β π΄ (π β π β§ π β π β§ π β€ (π β¨ π)))) |
7 | 6 | imp 408 | 1 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β π) β βπ β π΄ (π β π β§ π β π β§ π β€ (π β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 βwrex 3074 class class class wbr 5110 βcfv 6501 (class class class)co 7362 Basecbs 17090 lecple 17147 joincjn 18207 Atomscatm 37754 HLchlt 37841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-ov 7365 df-cvlat 37813 df-hlat 37842 |
This theorem is referenced by: hlsupr2 37879 atbtwnexOLDN 37939 atbtwnex 37940 cdlemb 38286 lhpexle2lem 38501 lhpexle3lem 38503 cdlemf1 39053 cdlemg35 39205 |
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