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Theorem dalem16 35283
Description: Lemma for dath 35340. The atoms 𝐷, 𝐸, and 𝐹 form a line of perspectivity. This is Desargue's Theorem for the special case where planes 𝑌 and 𝑍 are different. (Contributed by NM, 7-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalem16.m = (meet‘𝐾)
dalem16.o 𝑂 = (LPlanes‘𝐾)
dalem16.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem16.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem16.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem16.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dalem16.f 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
Assertion
Ref Expression
dalem16 ((𝜑𝑌𝑍) → 𝐹 (𝐷 𝐸))

Proof of Theorem dalem16
StepHypRef Expression
1 dalema.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
2 dalemc.l . . . 4 = (le‘𝐾)
3 dalemc.j . . . 4 = (join‘𝐾)
4 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem16.m . . . 4 = (meet‘𝐾)
6 dalem16.o . . . 4 𝑂 = (LPlanes‘𝐾)
7 dalem16.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
8 dalem16.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
9 eqid 2651 . . . 4 (𝑌 𝑍) = (𝑌 𝑍)
10 dalem16.f . . . 4 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 35279 . . 3 (𝜑𝐹 (𝑌 𝑍))
1211adantr 480 . 2 ((𝜑𝑌𝑍) → 𝐹 (𝑌 𝑍))
13 dalem16.d . . . . . 6 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
141, 2, 3, 4, 5, 6, 7, 8, 9, 13dalem10 35277 . . . . 5 (𝜑𝐷 (𝑌 𝑍))
15 dalem16.e . . . . . 6 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 35278 . . . . 5 (𝜑𝐸 (𝑌 𝑍))
171dalemkelat 35228 . . . . . 6 (𝜑𝐾 ∈ Lat)
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 35266 . . . . . . 7 (𝜑𝐷𝐴)
19 eqid 2651 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2019, 4atbase 34894 . . . . . . 7 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
2118, 20syl 17 . . . . . 6 (𝜑𝐷 ∈ (Base‘𝐾))
221, 2, 3, 4, 5, 6, 7, 8, 15dalemeea 35267 . . . . . . 7 (𝜑𝐸𝐴)
2319, 4atbase 34894 . . . . . . 7 (𝐸𝐴𝐸 ∈ (Base‘𝐾))
2422, 23syl 17 . . . . . 6 (𝜑𝐸 ∈ (Base‘𝐾))
251, 6dalemyeb 35253 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
261dalemzeo 35237 . . . . . . . 8 (𝜑𝑍𝑂)
2719, 6lplnbase 35138 . . . . . . . 8 (𝑍𝑂𝑍 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . . 7 (𝜑𝑍 ∈ (Base‘𝐾))
2919, 5latmcl 17099 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑍 ∈ (Base‘𝐾)) → (𝑌 𝑍) ∈ (Base‘𝐾))
3017, 25, 28, 29syl3anc 1366 . . . . . 6 (𝜑 → (𝑌 𝑍) ∈ (Base‘𝐾))
3119, 2, 3latjle12 17109 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ (𝑌 𝑍) ∈ (Base‘𝐾))) → ((𝐷 (𝑌 𝑍) ∧ 𝐸 (𝑌 𝑍)) ↔ (𝐷 𝐸) (𝑌 𝑍)))
3217, 21, 24, 30, 31syl13anc 1368 . . . . 5 (𝜑 → ((𝐷 (𝑌 𝑍) ∧ 𝐸 (𝑌 𝑍)) ↔ (𝐷 𝐸) (𝑌 𝑍)))
3314, 16, 32mpbi2and 976 . . . 4 (𝜑 → (𝐷 𝐸) (𝑌 𝑍))
3433adantr 480 . . 3 ((𝜑𝑌𝑍) → (𝐷 𝐸) (𝑌 𝑍))
351dalemkehl 35227 . . . . 5 (𝜑𝐾 ∈ HL)
3635adantr 480 . . . 4 ((𝜑𝑌𝑍) → 𝐾 ∈ HL)
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 35270 . . . . . 6 (𝜑𝐷𝐸)
38 eqid 2651 . . . . . . 7 (LLines‘𝐾) = (LLines‘𝐾)
393, 4, 38llni2 35116 . . . . . 6 (((𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴) ∧ 𝐷𝐸) → (𝐷 𝐸) ∈ (LLines‘𝐾))
4035, 18, 22, 37, 39syl31anc 1369 . . . . 5 (𝜑 → (𝐷 𝐸) ∈ (LLines‘𝐾))
4140adantr 480 . . . 4 ((𝜑𝑌𝑍) → (𝐷 𝐸) ∈ (LLines‘𝐾))
421, 2, 3, 4, 5, 38, 6, 7, 8, 9dalem15 35282 . . . 4 ((𝜑𝑌𝑍) → (𝑌 𝑍) ∈ (LLines‘𝐾))
432, 38llncmp 35126 . . . 4 ((𝐾 ∈ HL ∧ (𝐷 𝐸) ∈ (LLines‘𝐾) ∧ (𝑌 𝑍) ∈ (LLines‘𝐾)) → ((𝐷 𝐸) (𝑌 𝑍) ↔ (𝐷 𝐸) = (𝑌 𝑍)))
4436, 41, 42, 43syl3anc 1366 . . 3 ((𝜑𝑌𝑍) → ((𝐷 𝐸) (𝑌 𝑍) ↔ (𝐷 𝐸) = (𝑌 𝑍)))
4534, 44mpbid 222 . 2 ((𝜑𝑌𝑍) → (𝐷 𝐸) = (𝑌 𝑍))
4612, 45breqtrrd 4713 1 ((𝜑𝑌𝑍) → 𝐹 (𝐷 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  Atomscatm 34868  HLchlt 34955  LLinesclln 35095  LPlanesclpl 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104
This theorem is referenced by:  dalem63  35339
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