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| Mirrors > Home > MPE Home > Th. List > Mathboxes > llnexch2N | Structured version Visualization version GIF version | ||
| Description: Line exchange property (compare cvlatexch2 38865 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| llnexch.l | β’ β€ = (leβπΎ) |
| llnexch.j | β’ β¨ = (joinβπΎ) |
| llnexch.m | β’ β§ = (meetβπΎ) |
| llnexch.a | β’ π΄ = (AtomsβπΎ) |
| llnexch.n | β’ π = (LLinesβπΎ) |
| Ref | Expression |
|---|---|
| llnexch2N | β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β ((π β§ π) β€ π β (π β§ π) β€ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | llnexch.l | . . 3 β’ β€ = (leβπΎ) | |
| 2 | llnexch.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 3 | llnexch.m | . . 3 β’ β§ = (meetβπΎ) | |
| 4 | llnexch.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
| 5 | llnexch.n | . . 3 β’ π = (LLinesβπΎ) | |
| 6 | 1, 2, 3, 4, 5 | llnexchb2 39398 | . 2 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β ((π β§ π) β€ π β (π β§ π) = (π β§ π))) |
| 7 | hllat 38891 | . . . . 5 β’ (πΎ β HL β πΎ β Lat) | |
| 8 | 7 | 3ad2ant1 1130 | . . . 4 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β πΎ β Lat) |
| 9 | simp21 1203 | . . . . 5 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β π) | |
| 10 | eqid 2725 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
| 11 | 10, 5 | llnbase 39038 | . . . . 5 β’ (π β π β π β (BaseβπΎ)) |
| 12 | 9, 11 | syl 17 | . . . 4 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β (BaseβπΎ)) |
| 13 | simp22 1204 | . . . . 5 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β π) | |
| 14 | 10, 5 | llnbase 39038 | . . . . 5 β’ (π β π β π β (BaseβπΎ)) |
| 15 | 13, 14 | syl 17 | . . . 4 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β π β (BaseβπΎ)) |
| 16 | 10, 1, 3 | latmle2 18456 | . . . 4 β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β (π β§ π) β€ π) |
| 17 | 8, 12, 15, 16 | syl3anc 1368 | . . 3 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β (π β§ π) β€ π) |
| 18 | breq1 5146 | . . 3 β’ ((π β§ π) = (π β§ π) β ((π β§ π) β€ π β (π β§ π) β€ π)) | |
| 19 | 17, 18 | syl5ibcom 244 | . 2 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ β§ π β π)) β ((π β§ π) = (π β§ π) β (π β§ π) β€ π)) |
| 20 | 6, 19 | sylbid 239 | 1 β’ ((πΎ β HL β§ (π β π β§ π β π β§ π β π) β§ ((π β§ π) β π΄ 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 wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5143 βcfv 6543 (class class class)co 7416 Basecbs 17179 lecple 17239 joincjn 18302 meetcmee 18303 Latclat 18422 Atomscatm 38791 HLchlt 38878 LLinesclln 39020 |
| 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 |
| 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 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-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 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-psubsp 39032 df-pmap 39033 df-padd 39325 |
| This theorem is referenced by: (None) |
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