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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 2728 | . . . 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 38854 | . . 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 494 | . 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 406 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 βwrex 3067 class class class wbr 5148 βcfv 6548 (class class class)co 7420 Basecbs 17180 lecple 17240 joincjn 18303 Atomscatm 38735 HLchlt 38822 |
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-ext 2699 |
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-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-cvlat 38794 df-hlat 38823 |
This theorem is referenced by: hlsupr2 38860 atbtwnexOLDN 38920 atbtwnex 38921 cdlemb 39267 lhpexle2lem 39482 lhpexle3lem 39484 cdlemf1 40034 cdlemg35 40186 |
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