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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncnvat | Structured version Visualization version GIF version |
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.) |
Ref | Expression |
---|---|
ltrnel.l | β’ β€ = (leβπΎ) |
ltrnel.a | β’ π΄ = (AtomsβπΎ) |
ltrnel.h | β’ π» = (LHypβπΎ) |
ltrnel.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
ltrncnvat | β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΄) β (β‘πΉβπ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1135 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΄) β π β π΄) | |
2 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | ltrnel.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atbase 38672 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
5 | ltrnel.h | . . . 4 β’ π» = (LHypβπΎ) | |
6 | ltrnel.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
7 | 2, 3, 5, 6 | ltrncnvatb 39522 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β (BaseβπΎ)) β (π β π΄ β (β‘πΉβπ) β π΄)) |
8 | 4, 7 | syl3an3 1162 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΄) β (π β π΄ β (β‘πΉβπ) β π΄)) |
9 | 1, 8 | mpbid 231 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π΄) β (β‘πΉβπ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β‘ccnv 5668 βcfv 6537 Basecbs 17153 lecple 17213 Atomscatm 38646 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 |
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-plt 18295 df-glb 18312 df-p0 18390 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-hlat 38734 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 |
This theorem is referenced by: ltrncnvel 39526 ltrncnv 39530 ltrneq2 39532 cdlemg17h 40052 |
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