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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem45 | Structured version Visualization version GIF version |
Description: Lemma for dath 39241. Dummy center of perspectivity π is not on the line πΊπ». (Contributed by NM, 16-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)))) |
dalem44.m | β’ β§ = (meetβπΎ) |
dalem44.o | β’ π = (LPlanesβπΎ) |
dalem44.y | β’ π = ((π β¨ π) β¨ π ) |
dalem44.z | β’ π = ((π β¨ π) β¨ π) |
dalem44.g | β’ πΊ = ((π β¨ π) β§ (π β¨ π)) |
dalem44.h | β’ π» = ((π β¨ π) β§ (π β¨ π)) |
dalem44.i | β’ πΌ = ((π β¨ π ) β§ (π β¨ π)) |
Ref | Expression |
---|---|
dalem45 | β’ ((π β§ π = π β§ π) β Β¬ π β€ (πΊ β¨ π»)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.ph | . . . 4 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 39129 | . . 3 β’ (π β πΎ β Lat) |
3 | 2 | 3ad2ant1 1130 | . 2 β’ ((π β§ π = π β§ π) β πΎ β Lat) |
4 | dalem.ps | . . . 4 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)))) | |
5 | dalem.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 4, 5 | dalemcceb 39194 | . . 3 β’ (π β π β (BaseβπΎ)) |
7 | 6 | 3ad2ant3 1132 | . 2 β’ ((π β§ π = π β§ π) β π β (BaseβπΎ)) |
8 | 1 | dalemkehl 39128 | . . . 4 β’ (π β πΎ β HL) |
9 | 8 | 3ad2ant1 1130 | . . 3 β’ ((π β§ π = π β§ π) β πΎ β HL) |
10 | dalem.l | . . . 4 β’ β€ = (leβπΎ) | |
11 | dalem.j | . . . 4 β’ β¨ = (joinβπΎ) | |
12 | dalem44.m | . . . 4 β’ β§ = (meetβπΎ) | |
13 | dalem44.o | . . . 4 β’ π = (LPlanesβπΎ) | |
14 | dalem44.y | . . . 4 β’ π = ((π β¨ π) β¨ π ) | |
15 | dalem44.z | . . . 4 β’ π = ((π β¨ π) β¨ π) | |
16 | dalem44.g | . . . 4 β’ πΊ = ((π β¨ π) β§ (π β¨ π)) | |
17 | 1, 10, 11, 5, 4, 12, 13, 14, 15, 16 | dalem23 39201 | . . 3 β’ ((π β§ π = π β§ π) β πΊ β π΄) |
18 | dalem44.h | . . . 4 β’ π» = ((π β¨ π) β§ (π β¨ π)) | |
19 | 1, 10, 11, 5, 4, 12, 13, 14, 15, 18 | dalem29 39206 | . . 3 β’ ((π β§ π = π β§ π) β π» β π΄) |
20 | eqid 2728 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
21 | 20, 11, 5 | hlatjcl 38871 | . . 3 β’ ((πΎ β HL β§ πΊ β π΄ β§ π» β π΄) β (πΊ β¨ π») β (BaseβπΎ)) |
22 | 9, 17, 19, 21 | syl3anc 1368 | . 2 β’ ((π β§ π = π β§ π) β (πΊ β¨ π») β (BaseβπΎ)) |
23 | dalem44.i | . . . 4 β’ πΌ = ((π β¨ π ) β§ (π β¨ π)) | |
24 | 1, 10, 11, 5, 4, 12, 13, 14, 15, 23 | dalem34 39211 | . . 3 β’ ((π β§ π = π β§ π) β πΌ β π΄) |
25 | 20, 5 | atbase 38793 | . . 3 β’ (πΌ β π΄ β πΌ β (BaseβπΎ)) |
26 | 24, 25 | syl 17 | . 2 β’ ((π β§ π = π β§ π) β πΌ β (BaseβπΎ)) |
27 | 1, 10, 11, 5, 4, 12, 13, 14, 15, 16, 18, 23 | dalem44 39221 | . 2 β’ ((π β§ π = π β§ π) β Β¬ π β€ ((πΊ β¨ π») β¨ πΌ)) |
28 | 20, 10, 11 | latnlej2l 18459 | . 2 β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ (πΊ β¨ π») β (BaseβπΎ) β§ πΌ β (BaseβπΎ)) β§ Β¬ π β€ ((πΊ β¨ π») β¨ πΌ)) β Β¬ π β€ (πΊ β¨ π»)) |
29 | 3, 7, 22, 26, 27, 28 | syl131anc 1380 | 1 β’ ((π β§ π = π β§ π) β Β¬ π β€ (πΊ β¨ π»)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 meetcmee 18311 Latclat 18430 Atomscatm 38767 HLchlt 38854 LPlanesclpl 38997 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 |
This theorem is referenced by: dalem46 39223 dalem51 39228 dalem52 39229 |
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