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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnj | Structured version Visualization version GIF version | ||
| Description: Lattice translation of a meet. TODO: change antecedent to πΎ β HL (Contributed by NM, 25-May-2012.) |
| Ref | Expression |
|---|---|
| ltrnj.b | β’ π΅ = (BaseβπΎ) |
| ltrnj.j | β’ β¨ = (joinβπΎ) |
| ltrnj.h | β’ π» = (LHypβπΎ) |
| ltrnj.t | β’ π = ((LTrnβπΎ)βπ) |
| Ref | Expression |
|---|---|
| ltrnj | β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β (πΉβ(π β¨ π)) = ((πΉβπ) β¨ (πΉβπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1l 1196 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β πΎ β HL) | |
| 2 | 1 | hllatd 38698 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β πΎ β Lat) |
| 3 | ltrnj.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 4 | eqid 2731 | . . . 4 β’ (LAutβπΎ) = (LAutβπΎ) | |
| 5 | ltrnj.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
| 6 | 3, 4, 5 | ltrnlaut 39458 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ β (LAutβπΎ)) |
| 7 | 6 | 3adant3 1131 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β πΉ β (LAutβπΎ)) |
| 8 | simp3l 1200 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β π β π΅) | |
| 9 | simp3r 1201 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β π β π΅) | |
| 10 | ltrnj.b | . . 3 β’ π΅ = (BaseβπΎ) | |
| 11 | ltrnj.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 12 | 10, 11, 4 | lautj 39428 | . 2 β’ ((πΎ β Lat β§ (πΉ β (LAutβπΎ) β§ π β π΅ β§ π β π΅)) β (πΉβ(π β¨ π)) = ((πΉβπ) β¨ (πΉβπ))) |
| 13 | 2, 7, 8, 9, 12 | syl13anc 1371 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΅ β§ π β π΅)) β (πΉβ(π β¨ π)) = ((πΉβπ) β¨ (πΉβπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 Basecbs 17151 joincjn 18274 Latclat 18394 HLchlt 38684 LHypclh 39319 LAutclaut 39320 LTrncltrn 39436 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-proset 18258 df-poset 18276 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-lat 18395 df-atl 38632 df-cvlat 38656 df-hlat 38685 df-laut 39324 df-ldil 39439 df-ltrn 39440 |
| This theorem is referenced by: cdlemc2 39527 cdlemd2 39534 cdlemg2l 39938 cdlemg17h 40003 cdlemg17 40012 |
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