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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme25cl | Structured version Visualization version GIF version |
Description: Show closure of the unique element in cdleme25c 39529. (Contributed by NM, 2-Feb-2013.) |
Ref | Expression |
---|---|
cdleme24.b | β’ π΅ = (BaseβπΎ) |
cdleme24.l | β’ β€ = (leβπΎ) |
cdleme24.j | β’ β¨ = (joinβπΎ) |
cdleme24.m | β’ β§ = (meetβπΎ) |
cdleme24.a | β’ π΄ = (AtomsβπΎ) |
cdleme24.h | β’ π» = (LHypβπΎ) |
cdleme24.u | β’ π = ((π β¨ π) β§ π) |
cdleme24.f | β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
cdleme24.n | β’ π = ((π β¨ π) β§ (πΉ β¨ ((π β¨ π ) β§ π))) |
cdleme25cl.i | β’ πΌ = (β©π’ β π΅ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ (π β¨ π)) β π’ = π)) |
Ref | Expression |
---|---|
cdleme25cl | 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§ Β¬ π β€ π) β§ (π β π β§ π β€ (π β¨ π))) β πΌ β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme25cl.i | . 2 β’ πΌ = (β©π’ β π΅ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ (π β¨ π)) β π’ = π)) | |
2 | cdleme24.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
3 | cdleme24.l | . . . 4 β’ β€ = (leβπΎ) | |
4 | cdleme24.j | . . . 4 β’ β¨ = (joinβπΎ) | |
5 | cdleme24.m | . . . 4 β’ β§ = (meetβπΎ) | |
6 | cdleme24.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | cdleme24.h | . . . 4 β’ π» = (LHypβπΎ) | |
8 | cdleme24.u | . . . 4 β’ π = ((π β¨ π) β§ π) | |
9 | cdleme24.f | . . . 4 β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
10 | cdleme24.n | . . . 4 β’ π = ((π β¨ π) β§ (πΉ β¨ ((π β¨ π ) β§ π))) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme25c 39529 | . . 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§ Β¬ π β€ π) β§ (π β π β§ π β€ (π β¨ π))) β β!π’ β π΅ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ (π β¨ π)) β π’ = π)) |
12 | riotacl 7385 | . . 3 β’ (β!π’ β π΅ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ (π β¨ π)) β π’ = π) β (β©π’ β π΅ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ (π β¨ π)) β π’ = π)) β π΅) | |
13 | 11, 12 | syl 17 | . 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§ Β¬ π β€ π) β§ (π β π β§ π β€ (π β¨ π))) β (β©π’ β π΅ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ (π β¨ π)) β π’ = π)) β π΅) |
14 | 1, 13 | eqeltrid 2835 | 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§ Β¬ π β€ π) β§ (π β π β§ π β€ (π β¨ π))) β πΌ β π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 βwral 3059 β!wreu 3372 class class class wbr 5147 βcfv 6542 β©crio 7366 (class class class)co 7411 Basecbs 17148 lecple 17208 joincjn 18268 meetcmee 18269 Atomscatm 38436 HLchlt 38523 LHypclh 39158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 |
This theorem is referenced by: cdleme26e 39533 cdleme26eALTN 39535 cdleme26fALTN 39536 cdleme26f 39537 cdleme26f2ALTN 39538 cdleme26f2 39539 cdleme27cl 39540 cdlemefs27cl 39587 cdlemefs32sn1aw 39588 cdleme43fsv1snlem 39594 cdleme41sn3a 39607 cdleme40m 39641 cdleme40n 39642 |
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