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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme46fvaw | Structured version Visualization version GIF version |
Description: Show that (πΉβπ ) is an atom not under π when π is an atom not under π. (Contributed by NM, 18-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 |
---|---|
cdleme46fvaw | 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§ Β¬ π β€ π)) β ((πΉβπ ) β π΄ β§ Β¬ (πΉβπ ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef46.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | cdlemef46.l | . 2 β’ β€ = (leβπΎ) | |
3 | cdlemef46.j | . 2 β’ β¨ = (joinβπΎ) | |
4 | cdlemef46.m | . 2 β’ β§ = (meetβπΎ) | |
5 | cdlemef46.a | . 2 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemef46.h | . 2 β’ π» = (LHypβπΎ) | |
7 | cdlemef46.u | . 2 β’ π = ((π β¨ π) β§ π) | |
8 | vex 3467 | . . 3 β’ π β V | |
9 | cdlemef46.d | . . . 4 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
10 | eqid 2725 | . . . 4 β’ ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
11 | 9, 10 | cdleme31sc 39909 | . . 3 β’ (π β V β β¦π / π‘β¦π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
12 | 8, 11 | ax-mp 5 | . 2 β’ β¦π / π‘β¦π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
13 | cdlemefs46.e | . 2 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
14 | eqid 2725 | . 2 β’ (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)) = (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)) | |
15 | eqid 2725 | . 2 β’ 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·) = 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·) | |
16 | eqid 2725 | . 2 β’ (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)))) = (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)))) | |
17 | cdlemef46.f | . 2 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₯)) | |
18 | 1, 2, 3, 4, 5, 6, 7, 12, 9, 13, 14, 15, 16, 17 | cdleme32fvaw 39964 | 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§ Β¬ π β€ π)) β ((πΉβπ ) β π΄ β§ Β¬ (πΉβπ ) β€ 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 Vcvv 3463 β¦csb 3886 ifcif 4525 class class class wbr 5144 β¦ cmpt 5227 βcfv 6543 β©crio 7368 (class class class)co 7413 Basecbs 17174 lecple 17234 joincjn 18297 meetcmee 18298 Atomscatm 38787 HLchlt 38874 LHypclh 39509 |
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-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-riotaBAD 38477 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7987 df-2nd 7988 df-undef 8272 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-oposet 38700 df-ol 38702 df-oml 38703 df-covers 38790 df-ats 38791 df-atl 38822 df-cvlat 38846 df-hlat 38875 df-llines 39023 df-lplanes 39024 df-lvols 39025 df-lines 39026 df-psubsp 39028 df-pmap 39029 df-padd 39321 df-lhyp 39513 |
This theorem is referenced by: cdleme48bw 40027 cdleme48b 40028 cdlemeg46c 40038 cdlemeg46fvaw 40041 cdlemeg46frv 40050 cdlemeg46rgv 40053 cdlemeg46req 40054 cdlemeg46gfv 40055 cdleme48d 40060 cdlemg1fvawlemN 40098 |
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