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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 38672 | . . . 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 39778 | . . 3 β’ ((π β π΅ β§ π = π) β (πΉβπ) = π) |
8 | 5, 7 | sylan 579 | . 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 484 | . 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 2766 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β¦csb 3888 ifcif 4523 class class class wbr 5141 β¦ cmpt 5224 βcfv 6537 β©crio 7360 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 meetcmee 18277 Atomscatm 38646 HLchlt 38733 LHypclh 39368 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ats 38650 |
This theorem is referenced by: cdleme17d 39882 |
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