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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 1197 | . . 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 37932 | . 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 38692 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ β (LAutβπΎ)) |
| 7 | 6 | 3adant3 1132 | . 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 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΅) | |
| 9 | simp3r 1202 | . 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 38662 | . 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 1372 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6516 (class class class)co 7377 Basecbs 17109 joincjn 18229 Latclat 18349 HLchlt 37918 LHypclh 38553 LAutclaut 38554 LTrncltrn 38670 |
| 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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
| 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 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-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-map 8789 df-proset 18213 df-poset 18231 df-lub 18264 df-glb 18265 df-join 18266 df-meet 18267 df-lat 18350 df-atl 37866 df-cvlat 37890 df-hlat 37919 df-laut 38558 df-ldil 38673 df-ltrn 38674 |
| This theorem is referenced by: cdlemc2 38761 cdlemd2 38768 cdlemg2l 39172 cdlemg17h 39237 cdlemg17 39246 |
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