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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem18 | Structured version Visualization version GIF version |
Description: Lemma for dath 38602. 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 38489 | . . 3 β’ (π β πΎ β HL) |
3 | 1 | dalempea 38492 | . . 3 β’ (π β π β π΄) |
4 | 1 | dalemqea 38493 | . . 3 β’ (π β π β π΄) |
5 | 1 | dalemrea 38494 | . . 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 38335 | . . 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 1372 | . 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 5156 | . . . 4 β’ (π β€ π β π β€ ((π β¨ π) β¨ π )) |
13 | 12 | notbii 319 | . . 3 β’ (Β¬ π β€ π β Β¬ π β€ ((π β¨ π) β¨ π )) |
14 | 13 | rexbii 3094 | . 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwrex 3070 class class class wbr 5148 βcfv 6543 (class class class)co 7408 Basecbs 17143 lecple 17203 joincjn 18263 Atomscatm 38128 HLchlt 38215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 7364 df-ov 7411 df-oprab 7412 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 |
This theorem is referenced by: dalem20 38559 |
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