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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme17d4 | Structured version Visualization version GIF version |
Description: TODO: FIX COMMENT. (Contributed by NM, 11-Apr-2013.) |
Ref | Expression |
---|---|
cdlemef46.b | β’ π΅ = (BaseβπΎ) |
cdlemef46.l | β’ β€ = (leβπΎ) |
cdlemef46.j | β’ β¨ = (joinβπΎ) |
cdlemef46.m | β’ β§ = (meetβπΎ) |
cdlemef46.a | β’ π΄ = (AtomsβπΎ) |
cdlemef46.h | β’ π» = (LHypβπΎ) |
cdlemef46.u | β’ π = ((π β¨ π) β§ π) |
cdlemef46.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemefs46.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdlemef46.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₯)) |
Ref | Expression |
---|---|
cdleme17d4 | 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 = π) β (πΉβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2l 1196 | . . . 4 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΄) | |
2 | cdlemef46.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | cdlemef46.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atbase 38817 | . . . 4 β’ (π β π΄ β π β π΅) |
5 | 1, 4 | syl 17 | . . 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΅) |
6 | cdlemef46.f | . . . 4 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₯)) | |
7 | 6 | cdleme31id 39923 | . . 3 β’ ((π β π΅ β§ π = π) β (πΉβπ) = π) |
8 | 5, 7 | sylan 578 | . 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 = π) β (πΉβπ) = π) |
9 | simpr 483 | . 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 = π) β π = π) | |
10 | 8, 9 | eqtrd 2765 | 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 = π) β (πΉβπ) = 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 3884 ifcif 4524 class class class wbr 5143 β¦ cmpt 5226 βcfv 6543 β©crio 7371 (class class class)co 7416 Basecbs 17179 lecple 17239 joincjn 18302 meetcmee 18303 Atomscatm 38791 HLchlt 38878 LHypclh 39513 |
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-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 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-iota 6495 df-fun 6545 df-fv 6551 df-ats 38795 |
This theorem is referenced by: cdleme17d 40027 |
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