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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 39226. (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 39226 | . . 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 7383 | . . 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 2838 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 βwral 3062 β!wreu 3375 class class class wbr 5149 βcfv 6544 β©crio 7364 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 HLchlt 38220 LHypclh 38855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 df-lines 38372 df-psubsp 38374 df-pmap 38375 df-padd 38667 df-lhyp 38859 |
This theorem is referenced by: cdleme26e 39230 cdleme26eALTN 39232 cdleme26fALTN 39233 cdleme26f 39234 cdleme26f2ALTN 39235 cdleme26f2 39236 cdleme27cl 39237 cdlemefs27cl 39284 cdlemefs32sn1aw 39285 cdleme43fsv1snlem 39291 cdleme41sn3a 39304 cdleme40m 39338 cdleme40n 39339 |
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