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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2atnelpln | Structured version Visualization version GIF version |
Description: The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.) |
Ref | Expression |
---|---|
2atnelpln.j | β’ β¨ = (joinβπΎ) |
2atnelpln.a | β’ π΄ = (AtomsβπΎ) |
2atnelpln.p | β’ π = (LPlanesβπΎ) |
Ref | Expression |
---|---|
2atnelpln | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β Β¬ (π β¨ π ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38746 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | 3ad2ant1 1130 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β πΎ β Lat) |
3 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | 2atnelpln.j | . . . 4 β’ β¨ = (joinβπΎ) | |
5 | 2atnelpln.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 3, 4, 5 | hlatjcl 38750 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π ) β (BaseβπΎ)) |
7 | eqid 2726 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
8 | 3, 7 | latref 18406 | . . 3 β’ ((πΎ β Lat β§ (π β¨ π ) β (BaseβπΎ)) β (π β¨ π )(leβπΎ)(π β¨ π )) |
9 | 2, 6, 8 | syl2anc 583 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π )(leβπΎ)(π β¨ π )) |
10 | simpl1 1188 | . . . 4 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β¨ π ) β π) β πΎ β HL) | |
11 | simpr 484 | . . . 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) | |
12 | simpl2 1189 | . . . 4 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β¨ π ) β π) β π β π΄) | |
13 | simpl3 1190 | . . . 4 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β¨ π ) β π) β π β π΄) | |
14 | 2atnelpln.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
15 | 7, 4, 5, 14 | lplnnle2at 38925 | . . . 4 β’ ((πΎ β HL β§ ((π β¨ π ) β π β§ π β π΄ β§ π β π΄)) β Β¬ (π β¨ π )(leβπΎ)(π β¨ π )) |
16 | 10, 11, 12, 13, 15 | syl13anc 1369 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π β¨ π ) β π) β Β¬ (π β¨ π )(leβπΎ)(π β¨ π )) |
17 | 16 | ex 412 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β ((π β¨ π ) β π β Β¬ (π β¨ π )(leβπΎ)(π β¨ π ))) |
18 | 9, 17 | mt2d 136 | 1 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β 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 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 Latclat 18396 Atomscatm 38646 HLchlt 38733 LPlanesclpl 38876 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 |
This theorem is referenced by: islpln2a 38932 |
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