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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem62 | Structured version Visualization version GIF version |
Description: Lemma for dath 39119. Eliminate the condition π containing dummy variables π and π. (Contributed by NM, 11-Aug-2012.) |
Ref | Expression |
---|---|
dalem62.ph | β’ (π β (((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (π β π β§ π β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) |
dalem62.l | β’ β€ = (leβπΎ) |
dalem62.j | β’ β¨ = (joinβπΎ) |
dalem62.a | β’ π΄ = (AtomsβπΎ) |
dalem62.m | β’ β§ = (meetβπΎ) |
dalem62.o | β’ π = (LPlanesβπΎ) |
dalem62.y | β’ π = ((π β¨ π) β¨ π ) |
dalem62.z | β’ π = ((π β¨ π) β¨ π) |
dalem62.d | β’ π· = ((π β¨ π) β§ (π β¨ π)) |
dalem62.e | β’ πΈ = ((π β¨ π ) β§ (π β¨ π)) |
dalem62.f | β’ πΉ = ((π β¨ π) β§ (π β¨ π)) |
Ref | Expression |
---|---|
dalem62 | β’ ((π β§ π = π) β πΉ β€ (π· β¨ πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem62.ph | . . 3 β’ (π β (((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (π β π β§ π β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) | |
2 | dalem62.l | . . 3 β’ β€ = (leβπΎ) | |
3 | dalem62.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | dalem62.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | biid 261 | . . 3 β’ (((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π))) β ((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π)))) | |
6 | dalem62.o | . . 3 β’ π = (LPlanesβπΎ) | |
7 | dalem62.y | . . 3 β’ π = ((π β¨ π) β¨ π ) | |
8 | dalem62.z | . . 3 β’ π = ((π β¨ π) β¨ π) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | dalem20 39076 | . 2 β’ ((π β§ π = π) β βπβπ((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π)))) |
10 | dalem62.m | . . . . 5 β’ β§ = (meetβπΎ) | |
11 | dalem62.d | . . . . 5 β’ π· = ((π β¨ π) β§ (π β¨ π)) | |
12 | dalem62.e | . . . . 5 β’ πΈ = ((π β¨ π ) β§ (π β¨ π)) | |
13 | dalem62.f | . . . . 5 β’ πΉ = ((π β¨ π) β§ (π β¨ π)) | |
14 | 1, 2, 3, 4, 5, 10, 6, 7, 8, 11, 12, 13 | dalem61 39116 | . . . 4 β’ ((π β§ π = π β§ ((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π)))) β πΉ β€ (π· β¨ πΈ)) |
15 | 14 | 3expia 1118 | . . 3 β’ ((π β§ π = π) β (((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π))) β πΉ β€ (π· β¨ πΈ))) |
16 | 15 | exlimdvv 1929 | . 2 β’ ((π β§ π = π) β (βπβπ((π β π΄ β§ π β π΄) β§ Β¬ π β€ π β§ (π β π β§ Β¬ π β€ π β§ πΆ β€ (π β¨ π))) β πΉ β€ (π· β¨ πΈ))) |
17 | 9, 16 | mpd 15 | 1 β’ ((π β§ π = π) β πΉ β€ (π· β¨ πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 βwex 1773 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17150 lecple 17210 joincjn 18273 meetcmee 18274 Atomscatm 38645 HLchlt 38732 LPlanesclpl 38875 |
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 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 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-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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-lplanes 38882 df-lvols 38883 |
This theorem is referenced by: dalem63 39118 |
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