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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 2737 | . . 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 38602 | . 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 1204 | . . . 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 1138 | . . . 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 1202 | . . . 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 1205 | . . . 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 1203 | . . . 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 38635 | . . . 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 1380 | . . 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 3157 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwrex 3074 class class class wbr 5106 I cid 5531 βΎ cres 5636 βcfv 6497 Basecbs 17084 lecple 17141 Atomscatm 37728 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 |
This theorem is referenced by: ltrnnidn 38640 trlnidatb 38643 trlcone 39194 cdlemg46 39201 trljco 39206 cdlemh2 39282 cdlemh 39283 tendotr 39296 cdlemk3 39299 cdlemk12 39316 cdlemkole 39319 cdlemk14 39320 cdlemk15 39321 cdlemk1u 39325 cdlemk5u 39327 cdlemk12u 39338 cdlemk37 39380 cdlemk39 39382 cdlemkid1 39388 cdlemk47 39415 cdlemk51 39419 cdlemk52 39420 cdleml1N 39442 |
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