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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemeg46bOLDN | Structured version Visualization version GIF version |
Description: TODO FIX COMMENT. (Contributed by NM, 1-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemef46g.b | β’ π΅ = (BaseβπΎ) |
cdlemef46g.l | β’ β€ = (leβπΎ) |
cdlemef46g.j | β’ β¨ = (joinβπΎ) |
cdlemef46g.m | β’ β§ = (meetβπΎ) |
cdlemef46g.a | β’ π΄ = (AtomsβπΎ) |
cdlemef46g.h | β’ π» = (LHypβπΎ) |
cdlemef46g.u | β’ π = ((π β¨ π) β§ π) |
cdlemef46g.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemefs46g.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdlemef46g.f | β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) |
cdlemef46.v | β’ π = ((π β¨ π) β§ π) |
cdlemef46.n | β’ π = ((π£ β¨ π) β§ (π β¨ ((π β¨ π£) β§ π))) |
cdlemefs46.o | β’ π = ((π β¨ π) β§ (π β¨ ((π’ β¨ π£) β§ π))) |
cdlemef46.g | β’ πΊ = (π β π΅ β¦ if((π β π β§ Β¬ π β€ π), (β©π β π΅ βπ’ β π΄ ((Β¬ π’ β€ π β§ (π’ β¨ (π β§ π)) = π) β π = (if(π’ β€ (π β¨ π), (β©π β π΅ βπ£ β π΄ ((Β¬ π£ β€ π β§ Β¬ π£ β€ (π β¨ π)) β π = π)), β¦π’ / π£β¦π) β¨ (π β§ π)))), π)) |
Ref | Expression |
---|---|
cdlemeg46bOLDN | 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 β€ (π β¨ π)) β (πΊβπ) = β¦π / π£β¦π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef46g.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | cdlemef46g.l | . 2 β’ β€ = (leβπΎ) | |
3 | cdlemef46g.j | . 2 β’ β¨ = (joinβπΎ) | |
4 | cdlemef46g.m | . 2 β’ β§ = (meetβπΎ) | |
5 | cdlemef46g.a | . 2 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemef46g.h | . 2 β’ π» = (LHypβπΎ) | |
7 | cdlemef46.v | . 2 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemef46.n | . 2 β’ π = ((π£ β¨ π) β§ (π β¨ ((π β¨ π£) β§ π))) | |
9 | cdlemefs46.o | . 2 β’ π = ((π β¨ π) β§ (π β¨ ((π’ β¨ π£) β§ π))) | |
10 | cdlemef46.g | . 2 β’ πΊ = (π β π΅ β¦ if((π β π β§ Β¬ π β€ π), (β©π β π΅ βπ’ β π΄ ((Β¬ π’ β€ π β§ (π’ β¨ (π β§ π)) = π) β π = (if(π’ β€ (π β¨ π), (β©π β π΅ βπ£ β π΄ ((Β¬ π£ β€ π β§ Β¬ π£ β€ (π β¨ π)) β π = π)), β¦π’ / π£β¦π) β¨ (π β§ π)))), π)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdlemeg47b 38827 | 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 β€ (π β¨ π)) β (πΊβπ) = β¦π / π£β¦π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2940 βwral 3061 β¦csb 3843 ifcif 4474 class class class wbr 5093 β¦ cmpt 5176 βcfv 6480 β©crio 7293 (class class class)co 7338 Basecbs 17010 lecple 17067 joincjn 18127 meetcmee 18128 Atomscatm 37581 HLchlt 37668 LHypclh 38303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-iin 4945 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-1st 7900 df-2nd 7901 df-proset 18111 df-poset 18129 df-plt 18146 df-lub 18162 df-glb 18163 df-join 18164 df-meet 18165 df-p0 18241 df-p1 18242 df-lat 18248 df-clat 18315 df-oposet 37494 df-ol 37496 df-oml 37497 df-covers 37584 df-ats 37585 df-atl 37616 df-cvlat 37640 df-hlat 37669 df-lines 37820 df-psubsp 37822 df-pmap 37823 df-padd 38115 df-lhyp 38307 |
This theorem is referenced by: (None) |
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