Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrniotacl | Structured version Visualization version GIF version | ||
| Description: Version of cdleme50ltrn 39428 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 17-Apr-2013.) |
| Ref | Expression |
|---|---|
| ltrniotaval.l | β’ β€ = (leβπΎ) |
| ltrniotaval.a | β’ π΄ = (AtomsβπΎ) |
| ltrniotaval.h | β’ π» = (LHypβπΎ) |
| ltrniotaval.t | β’ π = ((LTrnβπΎ)βπ) |
| ltrniotaval.f | β’ πΉ = (β©π β π (πβπ) = π) |
| Ref | Expression |
|---|---|
| ltrniotacl | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2733 | . 2 β’ (BaseβπΎ) = (BaseβπΎ) | |
| 2 | ltrniotaval.l | . 2 β’ β€ = (leβπΎ) | |
| 3 | eqid 2733 | . 2 β’ (joinβπΎ) = (joinβπΎ) | |
| 4 | eqid 2733 | . 2 β’ (meetβπΎ) = (meetβπΎ) | |
| 5 | ltrniotaval.a | . 2 β’ π΄ = (AtomsβπΎ) | |
| 6 | ltrniotaval.h | . 2 β’ π» = (LHypβπΎ) | |
| 7 | eqid 2733 | . 2 β’ ((π(joinβπΎ)π)(meetβπΎ)π) = ((π(joinβπΎ)π)(meetβπΎ)π) | |
| 8 | eqid 2733 | . 2 β’ ((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π))) = ((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π))) | |
| 9 | eqid 2733 | . 2 β’ ((π(joinβπΎ)π)(meetβπΎ)(((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π)))(joinβπΎ)((π (joinβπΎ)π‘)(meetβπΎ)π))) = ((π(joinβπΎ)π)(meetβπΎ)(((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π)))(joinβπΎ)((π (joinβπΎ)π‘)(meetβπΎ)π))) | |
| 10 | eqid 2733 | . 2 β’ (π₯ β (BaseβπΎ) β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β (BaseβπΎ)βπ β π΄ ((Β¬ π β€ π β§ (π (joinβπΎ)(π₯(meetβπΎ)π)) = π₯) β π§ = (if(π β€ (π(joinβπΎ)π), (β©π¦ β (BaseβπΎ)βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π(joinβπΎ)π)) β π¦ = ((π(joinβπΎ)π)(meetβπΎ)(((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π)))(joinβπΎ)((π (joinβπΎ)π‘)(meetβπΎ)π))))), β¦π / π‘β¦((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π))))(joinβπΎ)(π₯(meetβπΎ)π)))), π₯)) = (π₯ β (BaseβπΎ) β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β (BaseβπΎ)βπ β π΄ ((Β¬ π β€ π β§ (π (joinβπΎ)(π₯(meetβπΎ)π)) = π₯) β π§ = (if(π β€ (π(joinβπΎ)π), (β©π¦ β (BaseβπΎ)βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π(joinβπΎ)π)) β π¦ = ((π(joinβπΎ)π)(meetβπΎ)(((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π)))(joinβπΎ)((π (joinβπΎ)π‘)(meetβπΎ)π))))), β¦π / π‘β¦((π‘(joinβπΎ)((π(joinβπΎ)π)(meetβπΎ)π))(meetβπΎ)(π(joinβπΎ)((π(joinβπΎ)π‘)(meetβπΎ)π))))(joinβπΎ)(π₯(meetβπΎ)π)))), π₯)) | |
| 11 | ltrniotaval.t | . 2 β’ π = ((LTrnβπΎ)βπ) | |
| 12 | ltrniotaval.f | . 2 β’ πΉ = (β©π β π (πβπ) = π) | |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemg1ltrnlem 39445 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ β π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 βwral 3062 β¦csb 3894 ifcif 4529 class class class wbr 5149 β¦ cmpt 5232 βcfv 6544 β©crio 7364 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 HLchlt 38220 LHypclh 38855 LTrncltrn 38972 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-riotaBAD 37823 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-undef 8258 df-map 8822 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 df-lines 38372 df-psubsp 38374 df-pmap 38375 df-padd 38667 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 |
| This theorem is referenced by: ltrniotacnvval 39453 ltrniotaidvalN 39454 ltrniotavalbN 39455 cdlemg1ci2 39457 cdlemki 39712 cdlemkj 39734 cdlemm10N 39989 dicssdvh 40057 dicvaddcl 40061 dicvscacl 40062 dicn0 40063 diclspsn 40065 cdlemn2 40066 cdlemn2a 40067 cdlemn3 40068 cdlemn4 40069 cdlemn4a 40070 cdlemn6 40073 cdlemn8 40075 cdlemn9 40076 cdlemn11a 40078 dihordlem7b 40086 dihopelvalcpre 40119 dih1 40157 dihmeetlem1N 40161 dihglblem5apreN 40162 dihglbcpreN 40171 dihmeetlem4preN 40177 dihmeetlem13N 40190 dih1dimatlem0 40199 dihatlat 40205 dihatexv 40209 dihjatcclem3 40291 dihjatcclem4 40292 |
| Copyright terms: Public domain | W3C validator |