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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem18 | Structured version Visualization version GIF version |
Description: Lemma for dath 38228. Show that a dummy atom π exists outside of the π and π planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-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) |
dalem18.y | β’ π = ((π β¨ π) β¨ π ) |
Ref | Expression |
---|---|
dalem18 | β’ (π β βπ β π΄ Β¬ π β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 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))))) | |
2 | 1 | dalemkehl 38115 | . . 3 β’ (π β πΎ β HL) |
3 | 1 | dalempea 38118 | . . 3 β’ (π β π β π΄) |
4 | 1 | dalemqea 38119 | . . 3 β’ (π β π β π΄) |
5 | 1 | dalemrea 38120 | . . 3 β’ (π β π β π΄) |
6 | dalemc.j | . . . 4 β’ β¨ = (joinβπΎ) | |
7 | dalemc.l | . . . 4 β’ β€ = (leβπΎ) | |
8 | dalemc.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
9 | 6, 7, 8 | 3dim3 37961 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β βπ β π΄ Β¬ π β€ ((π β¨ π) β¨ π )) |
10 | 2, 3, 4, 5, 9 | syl13anc 1373 | . 2 β’ (π β βπ β π΄ Β¬ π β€ ((π β¨ π) β¨ π )) |
11 | dalem18.y | . . . . 5 β’ π = ((π β¨ π) β¨ π ) | |
12 | 11 | breq2i 5118 | . . . 4 β’ (π β€ π β π β€ ((π β¨ π) β¨ π )) |
13 | 12 | notbii 320 | . . 3 β’ (Β¬ π β€ π β Β¬ π β€ ((π β¨ π) β¨ π )) |
14 | 13 | rexbii 3098 | . 2 β’ (βπ β π΄ Β¬ π β€ π β βπ β π΄ Β¬ π β€ ((π β¨ π) β¨ π )) |
15 | 10, 14 | sylibr 233 | 1 β’ (π β βπ 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 βwrex 3074 class class class wbr 5110 βcfv 6501 (class class class)co 7362 Basecbs 17090 lecple 17147 joincjn 18207 Atomscatm 37754 HLchlt 37841 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 |
This theorem is referenced by: dalem20 38185 |
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