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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlnidat | Structured version Visualization version GIF version |
Description: The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.) |
Ref | Expression |
---|---|
trlnidat.b | β’ π΅ = (BaseβπΎ) |
trlnidat.a | β’ π΄ = (AtomsβπΎ) |
trlnidat.h | β’ π» = (LHypβπΎ) |
trlnidat.t | β’ π = ((LTrnβπΎ)βπ) |
trlnidat.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
trlnidat | β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β (π βπΉ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlnidat.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2726 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
3 | trlnidat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | trlnidat.h | . . 3 β’ π» = (LHypβπΎ) | |
5 | trlnidat.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | ltrnnid 39520 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β βπ β π΄ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) |
7 | simp11 1200 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β§ π β π΄ β§ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) β (πΎ β HL β§ π β π»)) | |
8 | simp2 1134 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β§ π β π΄ β§ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) β π β π΄) | |
9 | simp3l 1198 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β§ π β π΄ β§ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) β Β¬ π(leβπΎ)π) | |
10 | simp12 1201 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β§ π β π΄ β§ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) β πΉ β π) | |
11 | simp3r 1199 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β§ π β π΄ β§ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) β (πΉβπ) β π) | |
12 | trlnidat.r | . . . . 5 β’ π = ((trLβπΎ)βπ) | |
13 | 2, 3, 4, 5, 12 | trlat 39553 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π(leβπΎ)π) β§ (πΉ β π β§ (πΉβπ) β π)) β (π βπΉ) β π΄) |
14 | 7, 8, 9, 10, 11, 13 | syl122anc 1376 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β§ π β π΄ β§ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π)) β (π βπΉ) β π΄) |
15 | 14 | rexlimdv3a 3153 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β (βπ β π΄ (Β¬ π(leβπΎ)π β§ (πΉβπ) β π) β (π βπΉ) β π΄)) |
16 | 6, 15 | mpd 15 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ πΉ β ( I βΎ π΅)) β (π βπΉ) 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 β wne 2934 βwrex 3064 class class class wbr 5141 I cid 5566 βΎ cres 5671 βcfv 6537 Basecbs 17153 lecple 17213 Atomscatm 38646 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 trLctrl 39542 |
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-mpo 7410 df-map 8824 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-p1 18391 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-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 |
This theorem is referenced by: ltrnnidn 39558 trlnidatb 39561 trlcone 40112 cdlemg46 40119 trljco 40124 cdlemh2 40200 cdlemh 40201 tendotr 40214 cdlemk3 40217 cdlemk12 40234 cdlemkole 40237 cdlemk14 40238 cdlemk15 40239 cdlemk1u 40243 cdlemk5u 40245 cdlemk12u 40256 cdlemk37 40298 cdlemk39 40300 cdlemkid1 40306 cdlemk47 40333 cdlemk51 40337 cdlemk52 40338 cdleml1N 40360 |
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