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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme32snb | Structured version Visualization version GIF version |
Description: Show closure of β¦π / π β¦π. (Contributed by NM, 1-Mar-2013.) |
Ref | Expression |
---|---|
cdleme32.b | β’ π΅ = (BaseβπΎ) |
cdleme32.l | β’ β€ = (leβπΎ) |
cdleme32.j | β’ β¨ = (joinβπΎ) |
cdleme32.m | β’ β§ = (meetβπΎ) |
cdleme32.a | β’ π΄ = (AtomsβπΎ) |
cdleme32.h | β’ π» = (LHypβπΎ) |
cdleme32.u | β’ π = ((π β¨ π) β§ π) |
cdleme32.c | β’ πΆ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
cdleme32.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdleme32.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdleme32.i | β’ πΌ = (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)) |
cdleme32.n | β’ π = if(π β€ (π β¨ π), πΌ, πΆ) |
Ref | Expression |
---|---|
cdleme32snb | 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π))) β β¦π / π β¦π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme32.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | cdleme32.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | cdleme32.j | . . . 4 β’ β¨ = (joinβπΎ) | |
4 | cdleme32.m | . . . 4 β’ β§ = (meetβπΎ) | |
5 | cdleme32.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | cdleme32.h | . . . 4 β’ π» = (LHypβπΎ) | |
7 | cdleme32.u | . . . 4 β’ π = ((π β¨ π) β§ π) | |
8 | cdleme32.c | . . . 4 β’ πΆ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
9 | cdleme32.d | . . . 4 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
10 | cdleme32.e | . . . 4 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
11 | cdleme32.i | . . . 4 β’ πΌ = (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)) | |
12 | cdleme32.n | . . . 4 β’ π = if(π β€ (π β¨ π), πΌ, πΆ) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdleme32snaw 39963 | . . 3 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π))) β (β¦π / π β¦π β π΄ β§ Β¬ β¦π / π β¦π β€ π)) |
14 | 13 | simpld 493 | . 2 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π))) β β¦π / π β¦π β π΄) |
15 | 1, 5 | atbase 38816 | . 2 β’ (β¦π / π β¦π β π΄ β β¦π / π β¦π β π΅) |
16 | 14, 15 | syl 17 | 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§ (π β π β§ (π β π΄ β§ Β¬ π β€ π))) β β¦π / π β¦π 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 βwral 3051 β¦csb 3885 ifcif 4524 class class class wbr 5143 βcfv 6542 β©crio 7370 (class class class)co 7415 Basecbs 17177 lecple 17237 joincjn 18300 meetcmee 18301 Atomscatm 38790 HLchlt 38877 LHypclh 39512 |
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-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 |
This theorem is referenced by: cdleme32fva 39965 |
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