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Mathbox for Norm Megill |
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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 38225 | . . . . 5 β’ (πΎ β HL β πΎ β AtLat) | |
2 | 1 | ad3antrrr 728 | . . . 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 38173 | . . . 4 β’ ((πΎ β AtLat β§ (π βπΉ) β π΄) β (π βπΉ) β 0 ) |
6 | 2, 5 | sylancom 588 | . . 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 413 | . 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 39037 | . . . 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 862 | . . 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 2957 | . 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 396 = wceq 1541 β wcel 2106 β wne 2940 βcfv 6543 0.cp0 18375 Atomscatm 38128 AtLatcal 38129 HLchlt 38215 LHypclh 38850 LTrncltrn 38967 trLctrl 39024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-map 8821 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-lhyp 38854 df-laut 38855 df-ldil 38970 df-ltrn 38971 df-trl 39025 |
This theorem is referenced by: trlid0b 39044 cdlemg12e 39513 trlcoat 39589 |
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