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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem15 | Structured version Visualization version GIF version |
Description: Lemma for dath 38607. The axis of perspectivity π is a line. (Contributed by NM, 21-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) |
dalem15.m | β’ β§ = (meetβπΎ) |
dalem15.n | β’ π = (LLinesβπΎ) |
dalem15.o | β’ π = (LPlanesβπΎ) |
dalem15.y | β’ π = ((π β¨ π) β¨ π ) |
dalem15.z | β’ π = ((π β¨ π) β¨ π) |
dalem15.x | β’ π = (π β§ π) |
Ref | Expression |
---|---|
dalem15 | β’ ((π β§ π β π) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem15.x | . 2 β’ π = (π β§ π) | |
2 | dalema.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))))) | |
3 | dalemc.l | . . . 4 β’ β€ = (leβπΎ) | |
4 | dalemc.j | . . . 4 β’ β¨ = (joinβπΎ) | |
5 | dalemc.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | dalem15.o | . . . 4 β’ π = (LPlanesβπΎ) | |
7 | eqid 2733 | . . . 4 β’ (LVolsβπΎ) = (LVolsβπΎ) | |
8 | dalem15.y | . . . 4 β’ π = ((π β¨ π) β¨ π ) | |
9 | dalem15.z | . . . 4 β’ π = ((π β¨ π) β¨ π) | |
10 | eqid 2733 | . . . 4 β’ (π β¨ πΆ) = (π β¨ πΆ) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | dalem14 38548 | . . 3 β’ ((π β§ π β π) β (π β¨ π) β (LVolsβπΎ)) |
12 | 2 | dalemkehl 38494 | . . . . 5 β’ (π β πΎ β HL) |
13 | 2 | dalemyeo 38503 | . . . . 5 β’ (π β π β π) |
14 | 2 | dalemzeo 38504 | . . . . 5 β’ (π β π β π) |
15 | dalem15.m | . . . . . 6 β’ β§ = (meetβπΎ) | |
16 | dalem15.n | . . . . . 6 β’ π = (LLinesβπΎ) | |
17 | 4, 15, 16, 6, 7 | 2lplnmj 38493 | . . . . 5 β’ ((πΎ β HL β§ π β π β§ π β π) β ((π β§ π) β π β (π β¨ π) β (LVolsβπΎ))) |
18 | 12, 13, 14, 17 | syl3anc 1372 | . . . 4 β’ (π β ((π β§ π) β π β (π β¨ π) β (LVolsβπΎ))) |
19 | 18 | adantr 482 | . . 3 β’ ((π β§ π β π) β ((π β§ π) β π β (π β¨ π) β (LVolsβπΎ))) |
20 | 11, 19 | mpbird 257 | . 2 β’ ((π β§ π β π) β (π β§ π) β π) |
21 | 1, 20 | eqeltrid 2838 | 1 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 HLchlt 38220 LLinesclln 38362 LPlanesclpl 38363 LVolsclvol 38364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 |
This theorem is referenced by: dalem16 38550 dalem53 38596 |
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