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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlexch2 | Structured version Visualization version GIF version |
Description: A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.) |
Ref | Expression |
---|---|
hlsuprexch.b | β’ π΅ = (BaseβπΎ) |
hlsuprexch.l | β’ β€ = (leβπΎ) |
hlsuprexch.j | β’ β¨ = (joinβπΎ) |
hlsuprexch.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlexch2 | β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΅) β§ Β¬ π β€ π) β (π β€ (π β¨ π) β π β€ (π β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 38863 | . 2 β’ (πΎ β HL β πΎ β CvLat) | |
2 | hlsuprexch.b | . . 3 β’ π΅ = (BaseβπΎ) | |
3 | hlsuprexch.l | . . 3 β’ β€ = (leβπΎ) | |
4 | hlsuprexch.j | . . 3 β’ β¨ = (joinβπΎ) | |
5 | hlsuprexch.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | 2, 3, 4, 5 | cvlexch2 38833 | . 2 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅) β§ Β¬ π β€ π) β (π β€ (π β¨ π) β π β€ (π β¨ π))) |
7 | 1, 6 | syl3an1 1160 | 1 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΅) β§ Β¬ π β€ 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§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 Atomscatm 38767 CvLatclc 38769 HLchlt 38854 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-lub 18345 df-join 18347 df-lat 18431 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 |
This theorem is referenced by: cdlemc3 39698 cdlemg4 40122 cdlemg6c 40125 |
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