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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlne | Structured version Visualization version GIF version |
Description: The trace of a lattice translation is not equal to any atom not under the fiducial co-atom π. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.) |
Ref | Expression |
---|---|
trlne.l | β’ β€ = (leβπΎ) |
trlne.a | β’ π΄ = (AtomsβπΎ) |
trlne.h | β’ π» = (LHypβπΎ) |
trlne.t | β’ π = ((LTrnβπΎ)βπ) |
trlne.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
trlne | β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β π β (π βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3r 1202 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β Β¬ π β€ π) | |
2 | trlne.l | . . . . . 6 β’ β€ = (leβπΎ) | |
3 | trlne.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
4 | trlne.t | . . . . . 6 β’ π = ((LTrnβπΎ)βπ) | |
5 | trlne.r | . . . . . 6 β’ π = ((trLβπΎ)βπ) | |
6 | 2, 3, 4, 5 | trlle 38237 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β€ π) |
7 | 6 | 3adant3 1132 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β (π βπΉ) β€ π) |
8 | breq1 5084 | . . . 4 β’ (π = (π βπΉ) β (π β€ π β (π βπΉ) β€ π)) | |
9 | 7, 8 | syl5ibrcom 248 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β (π = (π βπΉ) β π β€ π)) |
10 | 9 | necon3bd 2955 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β (Β¬ π β€ π β π β (π βπΉ))) |
11 | 1, 10 | mpd 15 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β π β (π βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1087 = wceq 1539 β wcel 2104 β wne 2941 class class class wbr 5081 βcfv 6454 lecple 17010 Atomscatm 37316 HLchlt 37403 LHypclh 38037 LTrncltrn 38154 trLctrl 38211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-map 8644 df-proset 18054 df-poset 18072 df-plt 18089 df-lub 18105 df-glb 18106 df-join 18107 df-meet 18108 df-p0 18184 df-p1 18185 df-lat 18191 df-oposet 37229 df-ol 37231 df-oml 37232 df-covers 37319 df-ats 37320 df-atl 37351 df-cvlat 37375 df-hlat 37404 df-lhyp 38041 df-laut 38042 df-ldil 38157 df-ltrn 38158 df-trl 38212 |
This theorem is referenced by: trlnle 38239 |
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