Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpmcvr | Structured version Visualization version GIF version | ||
| Description: The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.) |
| Ref | Expression |
|---|---|
| lhpmcvr.b | β’ π΅ = (BaseβπΎ) |
| lhpmcvr.l | β’ β€ = (leβπΎ) |
| lhpmcvr.m | β’ β§ = (meetβπΎ) |
| lhpmcvr.c | β’ πΆ = ( β βπΎ) |
| lhpmcvr.h | β’ π» = (LHypβπΎ) |
| Ref | Expression |
|---|---|
| lhpmcvr | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (π β§ π)πΆπ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hllat 38536 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
| 2 | 1 | ad2antrr 722 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β πΎ β Lat) |
| 3 | simprl 767 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β π β π΅) | |
| 4 | lhpmcvr.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 5 | lhpmcvr.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 6 | 4, 5 | lhpbase 39172 | . . . 4 β’ (π β π» β π β π΅) |
| 7 | 6 | ad2antlr 723 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β π β π΅) |
| 8 | lhpmcvr.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 9 | 4, 8 | latmcom 18420 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 10 | 2, 3, 7, 9 | syl3anc 1369 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (π β§ π) = (π β§ π)) |
| 11 | eqid 2730 | . . . . . 6 β’ (1.βπΎ) = (1.βπΎ) | |
| 12 | lhpmcvr.c | . . . . . 6 β’ πΆ = ( β βπΎ) | |
| 13 | 11, 12, 5 | lhp1cvr 39173 | . . . . 5 β’ ((πΎ β HL β§ π β π») β ππΆ(1.βπΎ)) |
| 14 | 13 | adantr 479 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β ππΆ(1.βπΎ)) |
| 15 | lhpmcvr.l | . . . . 5 β’ β€ = (leβπΎ) | |
| 16 | eqid 2730 | . . . . 5 β’ (joinβπΎ) = (joinβπΎ) | |
| 17 | 4, 15, 16, 11, 5 | lhpj1 39196 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (π(joinβπΎ)π) = (1.βπΎ)) |
| 18 | 14, 17 | breqtrrd 5175 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β ππΆ(π(joinβπΎ)π)) |
| 19 | simpll 763 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β πΎ β HL) | |
| 20 | 4, 16, 8, 12 | cvrexch 38594 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β ((π β§ π)πΆπ β ππΆ(π(joinβπΎ)π))) |
| 21 | 19, 7, 3, 20 | syl3anc 1369 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β ((π β§ π)πΆπ β ππΆ(π(joinβπΎ)π))) |
| 22 | 18, 21 | mpbird 256 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (π β§ π)πΆπ) |
| 23 | 10, 22 | eqbrtrd 5169 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (π β§ π)πΆπ) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 joincjn 18268 meetcmee 18269 1.cp1 18381 Latclat 18388 β ccvr 38435 HLchlt 38523 LHypclh 39158 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-lhyp 39162 |
| This theorem is referenced by: lhpmcvr2 39198 lhpm0atN 39203 |
| Copyright terms: Public domain | W3C validator |