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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg10b | Structured version Visualization version GIF version |
Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? (Contributed by NM, 4-May-2013.) |
Ref | Expression |
---|---|
cdlemg8.l | β’ β€ = (leβπΎ) |
cdlemg8.j | β’ β¨ = (joinβπΎ) |
cdlemg8.m | β’ β§ = (meetβπΎ) |
cdlemg8.a | β’ π΄ = (AtomsβπΎ) |
cdlemg8.h | β’ π» = (LHypβπΎ) |
cdlemg8.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
cdlemg10b | β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ πΉ β π) β (((πΉβπ) β¨ (πΉβπ)) β§ π) = ((π β¨ π) β§ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg8.h | . 2 β’ π» = (LHypβπΎ) | |
2 | cdlemg8.t | . 2 β’ π = ((LTrnβπΎ)βπ) | |
3 | cdlemg8.l | . 2 β’ β€ = (leβπΎ) | |
4 | cdlemg8.j | . 2 β’ β¨ = (joinβπΎ) | |
5 | cdlemg8.a | . 2 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemg8.m | . 2 β’ β§ = (meetβπΎ) | |
7 | eqid 2725 | . 2 β’ ((π β¨ π) β§ π) = ((π β¨ π) β§ π) | |
8 | 1, 2, 3, 4, 5, 6, 7 | cdlemg2m 40133 | 1 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ πΉ β π) β (((πΉβπ) β¨ (πΉβπ)) β§ π) = ((π β¨ π) β§ π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5143 βcfv 6543 (class class class)co 7416 lecple 17239 joincjn 18302 meetcmee 18303 Atomscatm 38791 HLchlt 38878 LHypclh 39513 LTrncltrn 39630 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-riotaBAD 38481 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-undef 8277 df-map 8845 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 |
This theorem is referenced by: cdlemg10c 40168 |
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