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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem42 | Structured version Visualization version GIF version |
Description: Lemma for dath 38012. Auxiliary atoms πΊπ»πΌ form a plane. (Contributed by NM, 4-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 |
---|---|
dalem42 | β’ ((π β§ π = π β§ π) β ((πΊ β¨ π») β¨ πΌ) 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 | dalemkehl 37899 | . . 3 β’ (π β πΎ β HL) |
3 | 2 | 3ad2ant1 1132 | . 2 β’ ((π β§ π = π β§ π) β πΎ β HL) |
4 | dalem.l | . . 3 β’ β€ = (leβπΎ) | |
5 | dalem.j | . . 3 β’ β¨ = (joinβπΎ) | |
6 | dalem.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
7 | dalem.ps | . . 3 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)))) | |
8 | dalem38.m | . . 3 β’ β§ = (meetβπΎ) | |
9 | dalem38.o | . . 3 β’ π = (LPlanesβπΎ) | |
10 | dalem38.y | . . 3 β’ π = ((π β¨ π) β¨ π ) | |
11 | dalem38.z | . . 3 β’ π = ((π β¨ π) β¨ π) | |
12 | dalem38.g | . . 3 β’ πΊ = ((π β¨ π) β§ (π β¨ π)) | |
13 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dalem23 37972 | . 2 β’ ((π β§ π = π β§ π) β πΊ β π΄) |
14 | dalem38.h | . . 3 β’ π» = ((π β¨ π) β§ (π β¨ π)) | |
15 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 14 | dalem29 37977 | . 2 β’ ((π β§ π = π β§ π) β π» β π΄) |
16 | dalem38.i | . . 3 β’ πΌ = ((π β¨ π ) β§ (π β¨ π)) | |
17 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 16 | dalem34 37982 | . 2 β’ ((π β§ π = π β§ π) β πΌ β π΄) |
18 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16 | dalem41 37989 | . 2 β’ ((π β§ π = π β§ π) β πΊ β π») |
19 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16 | dalem40 37988 | . 2 β’ ((π β§ π = π β§ π) β Β¬ πΌ β€ (πΊ β¨ π»)) |
20 | 4, 5, 6, 9 | lplni2 37813 | . 2 β’ ((πΎ β HL β§ (πΊ β π΄ β§ π» β π΄ β§ πΌ β π΄) β§ (πΊ β π» β§ Β¬ πΌ β€ (πΊ β¨ π»))) β ((πΊ β¨ π») β¨ πΌ) β π) |
21 | 3, 13, 15, 17, 18, 19, 20 | syl132anc 1387 | 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 396 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2940 class class class wbr 5092 βcfv 6479 (class class class)co 7337 Basecbs 17009 lecple 17066 joincjn 18126 meetcmee 18127 Atomscatm 37538 HLchlt 37625 LPlanesclpl 37768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-proset 18110 df-poset 18128 df-plt 18145 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-p0 18240 df-lat 18247 df-clat 18314 df-oposet 37451 df-ol 37453 df-oml 37454 df-covers 37541 df-ats 37542 df-atl 37573 df-cvlat 37597 df-hlat 37626 df-llines 37774 df-lplanes 37775 df-lvols 37776 |
This theorem is referenced by: dalem44 37992 dalem51 37999 dalem52 38000 dalem55 38003 |
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