![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > trlatn0 | Structured version Visualization version GIF version |
Description: The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.) |
Ref | Expression |
---|---|
trl0a.z | β’ 0 = (0.βπΎ) |
trl0a.a | β’ π΄ = (AtomsβπΎ) |
trl0a.h | β’ π» = (LHypβπΎ) |
trl0a.t | β’ π = ((LTrnβπΎ)βπ) |
trl0a.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
trlatn0 | β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((π βπΉ) β π΄ β (π βπΉ) β 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatl 38230 | . . . . 5 β’ (πΎ β HL β πΎ β AtLat) | |
2 | 1 | ad3antrrr 729 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π) β§ (π βπΉ) β π΄) β πΎ β AtLat) |
3 | trl0a.z | . . . . 5 β’ 0 = (0.βπΎ) | |
4 | trl0a.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | atn0 38178 | . . . 4 β’ ((πΎ β AtLat β§ (π βπΉ) β π΄) β (π βπΉ) β 0 ) |
6 | 2, 5 | sylancom 589 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π) β§ (π βπΉ) β π΄) β (π βπΉ) β 0 ) |
7 | 6 | ex 414 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((π βπΉ) β π΄ β (π βπΉ) β 0 )) |
8 | trl0a.h | . . . . 5 β’ π» = (LHypβπΎ) | |
9 | trl0a.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
10 | trl0a.r | . . . . 5 β’ π = ((trLβπΎ)βπ) | |
11 | 3, 4, 8, 9, 10 | trlator0 39042 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((π βπΉ) β π΄ β¨ (π βπΉ) = 0 )) |
12 | 11 | ord 863 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (Β¬ (π βπΉ) β π΄ β (π βπΉ) = 0 )) |
13 | 12 | necon1ad 2958 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((π βπΉ) β 0 β (π βπΉ) β π΄)) |
14 | 7, 13 | impbid 211 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((π βπΉ) β π΄ β (π βπΉ) β 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2941 βcfv 6544 0.cp0 18376 Atomscatm 38133 AtLatcal 38134 HLchlt 38220 LHypclh 38855 LTrncltrn 38972 trLctrl 39029 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-map 8822 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 |
This theorem is referenced by: trlid0b 39049 cdlemg12e 39518 trlcoat 39594 |
Copyright terms: Public domain | W3C validator |