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 39416 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 2732 | . 2 β’ (BaseβπΎ) = (BaseβπΎ) | |
| 2 | ltrniotaval.l | . 2 β’ β€ = (leβπΎ) | |
| 3 | eqid 2732 | . 2 β’ (joinβπΎ) = (joinβπΎ) | |
| 4 | eqid 2732 | . 2 β’ (meetβπΎ) = (meetβπΎ) | |
| 5 | ltrniotaval.a | . 2 β’ π΄ = (AtomsβπΎ) | |
| 6 | ltrniotaval.h | . 2 β’ π» = (LHypβπΎ) | |
| 7 | eqid 2732 | . 2 β’ ((π(joinβπΎ)π)(meetβπΎ)π) = ((π(joinβπΎ)π)(meetβπΎ)π) | |
| 8 | eqid 2732 | . 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 2732 | . 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 2732 | . 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 39433 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 β¦csb 3892 ifcif 4527 class class class wbr 5147 β¦ cmpt 5230 βcfv 6540 β©crio 7360 (class class class)co 7405 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 Atomscatm 38121 HLchlt 38208 LHypclh 38843 LTrncltrn 38960 |
| 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-riotaBAD 37811 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-iin 4999 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 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-undef 8254 df-map 8818 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 |
| This theorem is referenced by: ltrniotacnvval 39441 ltrniotaidvalN 39442 ltrniotavalbN 39443 cdlemg1ci2 39445 cdlemki 39700 cdlemkj 39722 cdlemm10N 39977 dicssdvh 40045 dicvaddcl 40049 dicvscacl 40050 dicn0 40051 diclspsn 40053 cdlemn2 40054 cdlemn2a 40055 cdlemn3 40056 cdlemn4 40057 cdlemn4a 40058 cdlemn6 40061 cdlemn8 40063 cdlemn9 40064 cdlemn11a 40066 dihordlem7b 40074 dihopelvalcpre 40107 dih1 40145 dihmeetlem1N 40149 dihglblem5apreN 40150 dihglbcpreN 40159 dihmeetlem4preN 40165 dihmeetlem13N 40178 dih1dimatlem0 40187 dihatlat 40193 dihatexv 40197 dihjatcclem3 40279 dihjatcclem4 40280 |
| Copyright terms: Public domain | W3C validator |