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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkoatnle | Structured version Visualization version GIF version |
Description: Utility lemma. (Contributed by NM, 2-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk1.b | β’ π΅ = (BaseβπΎ) |
cdlemk1.l | β’ β€ = (leβπΎ) |
cdlemk1.j | β’ β¨ = (joinβπΎ) |
cdlemk1.m | β’ β§ = (meetβπΎ) |
cdlemk1.a | β’ π΄ = (AtomsβπΎ) |
cdlemk1.h | β’ π» = (LHypβπΎ) |
cdlemk1.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemk1.r | β’ π = ((trLβπΎ)βπ) |
cdlemk1.s | β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) |
cdlemk1.o | β’ π = (πβπ·) |
Ref | Expression |
---|---|
cdlemkoatnle | β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp11 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΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πΎ β HL β§ π β π»)) | |
2 | cdlemk1.o | . . 3 β’ π = (πβπ·) | |
3 | cdlemk1.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
4 | cdlemk1.l | . . . 4 β’ β€ = (leβπΎ) | |
5 | cdlemk1.j | . . . 4 β’ β¨ = (joinβπΎ) | |
6 | cdlemk1.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | cdlemk1.h | . . . 4 β’ π» = (LHypβπΎ) | |
8 | cdlemk1.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
9 | cdlemk1.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
10 | cdlemk1.m | . . . 4 β’ β§ = (meetβπΎ) | |
11 | cdlemk1.s | . . . 4 β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | cdlemksel 40370 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ·) β π) |
13 | 2, 12 | eqeltrid 2829 | . 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΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β π β π) |
14 | simp22 1204 | . 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΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (π β π΄ β§ Β¬ π β€ π)) | |
15 | 4, 6, 7, 8 | ltrnel 39664 | . 2 β’ (((πΎ β HL β§ π β π») β§ π β π β§ (π β π΄ β§ Β¬ π β€ π)) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) |
16 | 1, 13, 14, 15 | syl3anc 1368 | 1 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5144 β¦ cmpt 5227 I cid 5570 β‘ccnv 5672 βΎ cres 5675 β ccom 5677 βcfv 6543 β©crio 7368 (class class class)co 7413 Basecbs 17174 lecple 17234 joincjn 18297 meetcmee 18298 Atomscatm 38787 HLchlt 38874 LHypclh 39509 LTrncltrn 39626 trLctrl 39683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-riotaBAD 38477 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7987 df-2nd 7988 df-undef 8272 df-map 8840 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-oposet 38700 df-ol 38702 df-oml 38703 df-covers 38790 df-ats 38791 df-atl 38822 df-cvlat 38846 df-hlat 38875 df-llines 39023 df-lplanes 39024 df-lvols 39025 df-lines 39026 df-psubsp 39028 df-pmap 39029 df-padd 39321 df-lhyp 39513 df-laut 39514 df-ldil 39629 df-ltrn 39630 df-trl 39684 |
This theorem is referenced by: cdlemk14 40379 cdlemk16a 40381 cdlemk1u 40384 cdlemk5u 40386 cdlemk6u 40387 cdlemk7u 40395 cdlemk12u 40397 cdlemkoatnle-2N 40400 |
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