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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkuv2-2 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 2, where sigma2 (p) is π, f2 is πΆ, and k2 is π. (Contributed by NM, 2-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk2.b | β’ π΅ = (BaseβπΎ) |
cdlemk2.l | β’ β€ = (leβπΎ) |
cdlemk2.j | β’ β¨ = (joinβπΎ) |
cdlemk2.m | β’ β§ = (meetβπΎ) |
cdlemk2.a | β’ π΄ = (AtomsβπΎ) |
cdlemk2.h | β’ π» = (LHypβπΎ) |
cdlemk2.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemk2.r | β’ π = ((trLβπΎ)βπ) |
cdlemk2.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)))))) |
cdlemk2.q | β’ π = (πβπΆ) |
cdlemk2.v | β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) |
Ref | Expression |
---|---|
cdlemkuv2-2 | β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΆ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΆ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemk2.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | cdlemk2.l | . 2 β’ β€ = (leβπΎ) | |
3 | cdlemk2.j | . 2 β’ β¨ = (joinβπΎ) | |
4 | cdlemk2.m | . 2 β’ β§ = (meetβπΎ) | |
5 | cdlemk2.a | . 2 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemk2.h | . 2 β’ π» = (LHypβπΎ) | |
7 | cdlemk2.t | . 2 β’ π = ((LTrnβπΎ)βπ) | |
8 | cdlemk2.r | . 2 β’ π = ((trLβπΎ)βπ) | |
9 | cdlemk2.s | . 2 β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) | |
10 | cdlemk2.q | . 2 β’ π = (πβπΆ) | |
11 | cdlemk2.v | . 2 β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | cdlemkuv2 40395 | 1 β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΆ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΆ))))) |
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 5143 β¦ cmpt 5226 I cid 5569 β‘ccnv 5671 βΎ cres 5674 β ccom 5676 βcfv 6542 β©crio 7370 (class class class)co 7415 Basecbs 17177 lecple 17237 joincjn 18300 meetcmee 18301 Atomscatm 38790 HLchlt 38877 LHypclh 39512 LTrncltrn 39629 trLctrl 39686 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-riotaBAD 38480 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-undef 8275 df-map 8843 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 |
This theorem is referenced by: cdlemk22 40421 |
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