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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem14 | Structured version Visualization version GIF version |
Description: Lemma for dath 39074. Planes π and π form a 3-dimensional space (when they are different). (Contributed by NM, 22-Jul-2012.) |
Ref | Expression |
---|---|
dalema.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))))) |
dalemc.l | β’ β€ = (leβπΎ) |
dalemc.j | β’ β¨ = (joinβπΎ) |
dalemc.a | β’ π΄ = (AtomsβπΎ) |
dalem14.o | β’ π = (LPlanesβπΎ) |
dalem14.v | β’ π = (LVolsβπΎ) |
dalem14.y | β’ π = ((π β¨ π) β¨ π ) |
dalem14.z | β’ π = ((π β¨ π) β¨ π) |
dalem14.w | β’ π = (π β¨ πΆ) |
Ref | Expression |
---|---|
dalem14 | β’ ((π β§ π β π) β (π β¨ π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.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 | dalemc.l | . . 3 β’ β€ = (leβπΎ) | |
3 | dalemc.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | dalemc.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | dalem14.o | . . 3 β’ π = (LPlanesβπΎ) | |
6 | dalem14.y | . . 3 β’ π = ((π β¨ π) β¨ π ) | |
7 | dalem14.z | . . 3 β’ π = ((π β¨ π) β¨ π) | |
8 | dalem14.w | . . 3 β’ π = (π β¨ πΆ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | dalem13 39014 | . 2 β’ ((π β§ π β π) β (π β¨ π) = π) |
10 | dalem14.v | . . 3 β’ π = (LVolsβπΎ) | |
11 | 1, 2, 3, 4, 5, 10, 6, 7, 8 | dalem9 39010 | . 2 β’ ((π β§ π β π) β π β π) |
12 | 9, 11 | eqeltrd 2832 | 1 β’ ((π β§ π β π) β (π β¨ π) β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 class class class wbr 5148 βcfv 6543 (class class class)co 7412 Basecbs 17151 lecple 17211 joincjn 18274 Atomscatm 38600 HLchlt 38687 LPlanesclpl 38830 LVolsclvol 38831 |
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 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 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 38513 df-ol 38515 df-oml 38516 df-covers 38603 df-ats 38604 df-atl 38635 df-cvlat 38659 df-hlat 38688 df-llines 38836 df-lplanes 38837 df-lvols 38838 |
This theorem is referenced by: dalem15 39016 |
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