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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 2726 | . . 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 3472 | . . . . 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 39768 | . . . . 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 4548 | . . 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 2726 | . . 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 39795 | . 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 1239 | . . 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 39768 | . . 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 2769 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 Vcvv 3468 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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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-rmo 3370 df-reu 3371 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-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 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-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 |
This theorem is referenced by: cdlemefr45 39811 |
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