![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefr44 | Structured version Visualization version GIF version |
Description: Value of f(r) when r is an atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefr45 instead? TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013.) |
Ref | Expression |
---|---|
cdlemef44.b | β’ π΅ = (BaseβπΎ) |
cdlemef44.l | β’ β€ = (leβπΎ) |
cdlemef44.j | β’ β¨ = (joinβπΎ) |
cdlemef44.m | β’ β§ = (meetβπΎ) |
cdlemef44.a | β’ π΄ = (AtomsβπΎ) |
cdlemef44.h | β’ π» = (LHypβπΎ) |
cdlemef44.u | β’ π = ((π β¨ π) β§ π) |
cdlemef44.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemef44.o | 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), πΌ, β¦π / π‘β¦π·) β¨ (π₯ β§ π)))) |
cdlemef44.f | β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), π, π₯)) |
Ref | Expression |
---|---|
cdlemefr44 | 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 (πΉβπ ) = β¦π / π‘β¦π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef44.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | cdlemef44.l | . . 3 β’ β€ = (leβπΎ) | |
3 | cdlemef44.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | cdlemef44.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | cdlemef44.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemef44.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | cdlemef44.u | . . 3 β’ π = ((π β¨ π) β§ π) | |
8 | eqid 2737 | . . 3 β’ ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
9 | biid 261 | . . . 4 β’ (π β€ (π β¨ π) β π β€ (π β¨ π)) | |
10 | vex 3452 | . . . . 5 β’ π β V | |
11 | cdlemef44.d | . . . . . 6 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
12 | 11, 8 | cdleme31sc 38876 | . . . . 5 β’ (π β V β β¦π / π‘β¦π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
13 | 10, 12 | ax-mp 5 | . . . 4 β’ β¦π / π‘β¦π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
14 | 9, 13 | ifbieq2i 4516 | . . 3 β’ if(π β€ (π β¨ π), πΌ, β¦π / π‘β¦π·) = if(π β€ (π β¨ π), πΌ, ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
15 | cdlemef44.o | . . 3 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), πΌ, β¦π / π‘β¦π·) β¨ (π₯ β§ π)))) | |
16 | cdlemef44.f | . . 3 β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), π, π₯)) | |
17 | eqid 2737 | . . 3 β’ ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17 | cdlemefr31fv1 38903 | . 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§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β (πΉβπ ) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
19 | simp2rl 1243 | . . 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§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β π β π΄) | |
20 | 11, 17 | cdleme31sc 38876 | . . 3 β’ (π β π΄ β β¦π / π‘β¦π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
21 | 19, 20 | 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§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β β¦π / π‘β¦π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
22 | 18, 21 | eqtr4d 2780 | 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 (πΉβπ ) = β¦π / π‘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 2944 βwral 3065 Vcvv 3448 β¦csb 3860 ifcif 4491 class class class wbr 5110 β¦ cmpt 5193 βcfv 6501 β©crio 7317 (class class class)co 7362 Basecbs 17090 lecple 17147 joincjn 18207 meetcmee 18208 Atomscatm 37754 HLchlt 37841 LHypclh 38476 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-lines 37993 df-psubsp 37995 df-pmap 37996 df-padd 38288 df-lhyp 38480 |
This theorem is referenced by: cdlemefr45 38919 |
Copyright terms: Public domain | W3C validator |