![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 3473 | . . 3 β’ π β V | |
9 | cdlemef46.d | . . . 4 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
10 | eqid 2727 | . . . 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 39781 | . . 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 2727 | . 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 2727 | . 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 2727 | . 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 39836 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 βwral 3056 Vcvv 3469 β¦csb 3889 ifcif 4524 class class class wbr 5142 β¦ cmpt 5225 βcfv 6542 β©crio 7369 (class class class)co 7414 Basecbs 17165 lecple 17225 joincjn 18288 meetcmee 18289 Atomscatm 38659 HLchlt 38746 LHypclh 39381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-riotaBAD 38349 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-undef 8270 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-p1 18403 df-lat 18409 df-clat 18476 df-oposet 38572 df-ol 38574 df-oml 38575 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 df-llines 38895 df-lplanes 38896 df-lvols 38897 df-lines 38898 df-psubsp 38900 df-pmap 38901 df-padd 39193 df-lhyp 39385 |
This theorem is referenced by: cdleme48bw 39899 cdleme48b 39900 cdlemeg46c 39910 cdlemeg46fvaw 39913 cdlemeg46frv 39922 cdlemeg46rgv 39925 cdlemeg46req 39926 cdlemeg46gfv 39927 cdleme48d 39932 cdlemg1fvawlemN 39970 |
Copyright terms: Public domain | W3C validator |