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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 2724 | . . . 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 38756 | . . 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 1084 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 βwrex 3062 class class class wbr 5139 βcfv 6534 (class class class)co 7402 Basecbs 17149 lecple 17209 joincjn 18272 Atomscatm 38637 HLchlt 38724 |
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-ext 2695 |
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-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-ov 7405 df-cvlat 38696 df-hlat 38725 |
This theorem is referenced by: hlsupr2 38762 atbtwnexOLDN 38822 atbtwnex 38823 cdlemb 39169 lhpexle2lem 39384 lhpexle3lem 39386 cdlemf1 39936 cdlemg35 40088 |
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