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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlid0b | Structured version Visualization version GIF version |
Description: A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.) |
Ref | Expression |
---|---|
trlid0b.b | β’ π΅ = (BaseβπΎ) |
trlid0b.z | β’ 0 = (0.βπΎ) |
trlid0b.h | β’ π» = (LHypβπΎ) |
trlid0b.t | β’ π = ((LTrnβπΎ)βπ) |
trlid0b.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
trlid0b | β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΉ = ( I βΎ π΅) β (π βπΉ) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlid0b.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2726 | . . . 4 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | trlid0b.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | trlid0b.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
5 | trlid0b.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | trlnidatb 39560 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΉ β ( I βΎ π΅) β (π βπΉ) β (AtomsβπΎ))) |
7 | trlid0b.z | . . . 4 β’ 0 = (0.βπΎ) | |
8 | 7, 2, 3, 4, 5 | trlatn0 39555 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((π βπΉ) β (AtomsβπΎ) β (π βπΉ) β 0 )) |
9 | 6, 8 | bitrd 279 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΉ β ( I βΎ π΅) β (π βπΉ) β 0 )) |
10 | 9 | necon4bid 2980 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΉ = ( I βΎ π΅) β (π βπΉ) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 I cid 5566 βΎ cres 5671 βcfv 6536 Basecbs 17150 0.cp0 18385 Atomscatm 38645 HLchlt 38732 LHypclh 39367 LTrncltrn 39484 trLctrl 39541 |
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 7721 |
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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 |
This theorem is referenced by: trlnid 39562 trlcoat 40106 trlcone 40111 trljco 40123 tendoid 40156 tendoex 40358 dia0 40435 |
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