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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem43 | Structured version Visualization version GIF version |
Description: Lemma for dath 39118. Planes πΊπ»πΌ and π are different. (Contributed by NM, 8-Aug-2012.) |
Ref | Expression |
---|---|
dalem.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))))) |
dalem.l | β’ β€ = (leβπΎ) |
dalem.j | β’ β¨ = (joinβπΎ) |
dalem.a | β’ π΄ = (AtomsβπΎ) |
dalem.ps | 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)))) |
dalem38.m | β’ β§ = (meetβπΎ) |
dalem38.o | β’ π = (LPlanesβπΎ) |
dalem38.y | β’ π = ((π β¨ π) β¨ π ) |
dalem38.z | β’ π = ((π β¨ π) β¨ π) |
dalem38.g | β’ πΊ = ((π β¨ π) β§ (π β¨ π)) |
dalem38.h | β’ π» = ((π β¨ π) β§ (π β¨ π)) |
dalem38.i | β’ πΌ = ((π β¨ π ) β§ (π β¨ π)) |
Ref | Expression |
---|---|
dalem43 | β’ ((π β§ π = π β§ π) β ((πΊ β¨ π») β¨ πΌ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.ph | . . . . 5 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 | 1 | dalemkelat 39006 | . . . 4 β’ (π β πΎ β Lat) |
3 | 2 | 3ad2ant1 1130 | . . 3 β’ ((π β§ π = π β§ π) β πΎ β Lat) |
4 | 1 | dalemkehl 39005 | . . . . 5 β’ (π β πΎ β HL) |
5 | 4 | 3ad2ant1 1130 | . . . 4 β’ ((π β§ π = π β§ π) β πΎ β HL) |
6 | dalem.l | . . . . 5 β’ β€ = (leβπΎ) | |
7 | dalem.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
8 | dalem.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
9 | dalem.ps | . . . . 5 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)))) | |
10 | dalem38.m | . . . . 5 β’ β§ = (meetβπΎ) | |
11 | dalem38.o | . . . . 5 β’ π = (LPlanesβπΎ) | |
12 | dalem38.y | . . . . 5 β’ π = ((π β¨ π) β¨ π ) | |
13 | dalem38.z | . . . . 5 β’ π = ((π β¨ π) β¨ π) | |
14 | dalem38.g | . . . . 5 β’ πΊ = ((π β¨ π) β§ (π β¨ π)) | |
15 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 14 | dalem23 39078 | . . . 4 β’ ((π β§ π = π β§ π) β πΊ β π΄) |
16 | dalem38.h | . . . . 5 β’ π» = ((π β¨ π) β§ (π β¨ π)) | |
17 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 16 | dalem29 39083 | . . . 4 β’ ((π β§ π = π β§ π) β π» β π΄) |
18 | eqid 2726 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
19 | 18, 7, 8 | hlatjcl 38748 | . . . 4 β’ ((πΎ β HL β§ πΊ β π΄ β§ π» β π΄) β (πΊ β¨ π») β (BaseβπΎ)) |
20 | 5, 15, 17, 19 | syl3anc 1368 | . . 3 β’ ((π β§ π = π β§ π) β (πΊ β¨ π») β (BaseβπΎ)) |
21 | dalem38.i | . . . . 5 β’ πΌ = ((π β¨ π ) β§ (π β¨ π)) | |
22 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 21 | dalem34 39088 | . . . 4 β’ ((π β§ π = π β§ π) β πΌ β π΄) |
23 | 18, 8 | atbase 38670 | . . . 4 β’ (πΌ β π΄ β πΌ β (BaseβπΎ)) |
24 | 22, 23 | syl 17 | . . 3 β’ ((π β§ π = π β§ π) β πΌ β (BaseβπΎ)) |
25 | 18, 6, 7 | latlej2 18412 | . . 3 β’ ((πΎ β Lat β§ (πΊ β¨ π») β (BaseβπΎ) β§ πΌ β (BaseβπΎ)) β πΌ β€ ((πΊ β¨ π») β¨ πΌ)) |
26 | 3, 20, 24, 25 | syl3anc 1368 | . 2 β’ ((π β§ π = π β§ π) β πΌ β€ ((πΊ β¨ π») β¨ πΌ)) |
27 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 21 | dalem35 39089 | . 2 β’ ((π β§ π = π β§ π) β Β¬ πΌ β€ π) |
28 | nbrne1 5160 | . 2 β’ ((πΌ β€ ((πΊ β¨ π») β¨ πΌ) β§ Β¬ πΌ β€ π) β ((πΊ β¨ π») β¨ πΌ) β π) | |
29 | 26, 27, 28 | syl2anc 583 | 1 β’ ((π β§ π = π β§ π) β ((πΊ β¨ π») β¨ πΌ) β π) |
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 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17151 lecple 17211 joincjn 18274 meetcmee 18275 Latclat 18394 Atomscatm 38644 HLchlt 38731 LPlanesclpl 38874 |
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-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 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-lat 18395 df-clat 18462 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 |
This theorem is referenced by: dalem44 39098 dalem51 39105 |
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