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Theorem arglem1N 39517
Description: Lemma for Desargues's law. Theorem 13.3 of [Crawley] p. 110, third and fourth lines from bottom. In these lemmas, 𝑃, 𝑄, 𝑅, 𝑆, 𝑇, π‘ˆ, 𝐢, 𝐷, 𝐸, 𝐹, and 𝐺 represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
arglem1.j ∨ = (joinβ€˜πΎ)
arglem1.m ∧ = (meetβ€˜πΎ)
arglem1.a 𝐴 = (Atomsβ€˜πΎ)
arglem1.f 𝐹 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
arglem1.g 𝐺 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))
Assertion
Ref Expression
arglem1N ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝐹 ∈ 𝐴)

Proof of Theorem arglem1N
StepHypRef Expression
1 arglem1.f . 2 𝐹 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
2 simpl11 1245 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝐾 ∈ HL)
32hllatd 38690 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
4 simpl12 1246 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑃 ∈ 𝐴)
5 eqid 2724 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
6 arglem1.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
75, 6atbase 38615 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
84, 7syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
9 simpl13 1247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑄 ∈ 𝐴)
105, 6atbase 38615 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
119, 10syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
12 simpl21 1248 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑆 ∈ 𝐴)
135, 6atbase 38615 . . . . . 6 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
1412, 13syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑆 ∈ (Baseβ€˜πΎ))
15 simpl22 1249 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑇 ∈ 𝐴)
165, 6atbase 38615 . . . . . 6 (𝑇 ∈ 𝐴 β†’ 𝑇 ∈ (Baseβ€˜πΎ))
1715, 16syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑇 ∈ (Baseβ€˜πΎ))
18 arglem1.j . . . . . 6 ∨ = (joinβ€˜πΎ)
195, 18latj4 18443 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) ∧ (𝑆 ∈ (Baseβ€˜πΎ) ∧ 𝑇 ∈ (Baseβ€˜πΎ))) β†’ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) = ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)))
203, 8, 11, 14, 17, 19syl122anc 1376 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) = ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)))
21 arglem1.g . . . . . 6 𝐺 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))
22 simpr 484 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝐺 ∈ 𝐴)
2321, 22eqeltrrid 2830 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
24 simpl31 1251 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑃 β‰  𝑆)
25 eqid 2724 . . . . . . . 8 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
2618, 6, 25llni2 38839 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 β‰  𝑆) β†’ (𝑃 ∨ 𝑆) ∈ (LLinesβ€˜πΎ))
272, 4, 12, 24, 26syl31anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ (𝑃 ∨ 𝑆) ∈ (LLinesβ€˜πΎ))
28 simpl32 1252 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑄 β‰  𝑇)
2918, 6, 25llni2 38839 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 β‰  𝑇) β†’ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
302, 9, 15, 28, 29syl31anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
31 arglem1.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
32 eqid 2724 . . . . . . 7 (LPlanesβ€˜πΎ) = (LPlanesβ€˜πΎ)
3318, 31, 6, 25, 322llnmj 38887 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑆) ∈ (LLinesβ€˜πΎ) ∧ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ)) β†’ (((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)) ∈ (LPlanesβ€˜πΎ)))
342, 27, 30, 33syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ (((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)) ∈ (LPlanesβ€˜πΎ)))
3523, 34mpbid 231 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑆) ∨ (𝑄 ∨ 𝑇)) ∈ (LPlanesβ€˜πΎ))
3620, 35eqeltrd 2825 . . 3 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ (LPlanesβ€˜πΎ))
37 simpl23 1250 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑃 β‰  𝑄)
3818, 6, 25llni2 38839 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (𝑃 ∨ 𝑄) ∈ (LLinesβ€˜πΎ))
392, 4, 9, 37, 38syl31anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ (LLinesβ€˜πΎ))
40 simpl33 1253 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝑆 β‰  𝑇)
4118, 6, 25llni2 38839 . . . . 5 (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑆 β‰  𝑇) β†’ (𝑆 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
422, 12, 15, 40, 41syl31anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ (𝑆 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
4318, 31, 6, 25, 322llnmj 38887 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (LLinesβ€˜πΎ) ∧ (𝑆 ∨ 𝑇) ∈ (LLinesβ€˜πΎ)) β†’ (((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ (LPlanesβ€˜πΎ)))
442, 39, 42, 43syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ (((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴 ↔ ((𝑃 ∨ 𝑄) ∨ (𝑆 ∨ 𝑇)) ∈ (LPlanesβ€˜πΎ)))
4536, 44mpbird 257 . 2 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ 𝐴)
461, 45eqeltrid 2829 1 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝐹 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  β€˜cfv 6533  (class class class)co 7401  Basecbs 17142  joincjn 18265  meetcmee 18266  Latclat 18385  Atomscatm 38589  HLchlt 38676  LLinesclln 38818  LPlanesclpl 38819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-proset 18249  df-poset 18267  df-plt 18284  df-lub 18300  df-glb 18301  df-join 18302  df-meet 18303  df-p0 18379  df-lat 18386  df-clat 18453  df-oposet 38502  df-ol 38504  df-oml 38505  df-covers 38592  df-ats 38593  df-atl 38624  df-cvlat 38648  df-hlat 38677  df-llines 38825  df-lplanes 38826
This theorem is referenced by: (None)
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