Theorem cdleme16d 36810

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence. 𝐹 and 𝐺 represent f(s) and f(t) respectively. We show, in their notation, (s ∨ t) ∧ (f(s) ∨ f(t)) is an atom. (Contributed by NM, 11-Oct-2012.)

Hypotheses

Ref	Expression
cdleme12.l	⊢ ≤ = (le‘𝐾)
cdleme12.j	⊢ ∨ = (join‘𝐾)
cdleme12.m	⊢ ∧ = (meet‘𝐾)
cdleme12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme12.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme12.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme12.g	⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme16d	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) ∈ 𝐴)

Proof of Theorem cdleme16d

Step	Hyp	Ref	Expression
1		cdleme12.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		cdleme12.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		cdleme12.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
4		cdleme12.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdleme12.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdleme12.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
7		cdleme12.f	. . . 4 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
8		cdleme12.g	. . . 4 ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
9	1, 2, 3, 4, 5, 6, 7, 8	cdleme16c 36809	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∨ (𝐹 ∨ 𝐺)) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
10		simp23r 1275	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ≠ 𝑇)
11		simp33 1191	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))
12		simp11l 1264	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ HL)
13		simp21l 1270	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ∈ 𝐴)
14		simp22l 1272	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑇 ∈ 𝐴)
15		simp11r 1265	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑊 ∈ 𝐻)
16		simp12l 1266	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ∈ 𝐴)
17		simp12r 1267	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑃 ≤ 𝑊)
18		simp13l 1268	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑄 ∈ 𝐴)
19		simp23l 1274	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ≠ 𝑄)
20	1, 2, 3, 4, 5, 6	lhpat2 36574	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
21	12, 15, 16, 17, 18, 19, 20	syl222anc 1366	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑈 ∈ 𝐴)
22		eqid 2772	. . . . . 6 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾)
23	1, 2, 4, 22	islpln2a 36077	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (LPlanes‘𝐾) ↔ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))))
24	12, 13, 14, 21, 23	syl13anc 1352	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (LPlanes‘𝐾) ↔ (𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))))
25	10, 11, 24	mpbir2and 700	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (LPlanes‘𝐾))
26	9, 25	eqeltrd 2860	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∨ (𝐹 ∨ 𝐺)) ∈ (LPlanes‘𝐾))
27		eqid 2772	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
28	2, 4, 27	islln2a 36046	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → ((𝑆 ∨ 𝑇) ∈ (LLines‘𝐾) ↔ 𝑆 ≠ 𝑇))
29	12, 13, 14, 28	syl3anc 1351	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∈ (LLines‘𝐾) ↔ 𝑆 ≠ 𝑇))
30	10, 29	mpbird 249	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾))
31	1, 2, 3, 4, 5, 6, 7, 8	cdleme16b 36808	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐹 ≠ 𝐺)
32		simp11 1183	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simp12 1184	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34		simp13 1185	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
35		simp21 1186	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
36		simp31 1189	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
37	1, 2, 3, 4, 5, 6, 7	cdleme3fa 36765	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝐴)
38	32, 33, 34, 35, 19, 36, 37	syl132anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐹 ∈ 𝐴)
39		simp22 1187	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
40		simp32 1190	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
41	1, 2, 3, 4, 5, 6, 8	cdleme3fa 36765	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ 𝐴)
42	32, 33, 34, 39, 19, 40, 41	syl132anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐺 ∈ 𝐴)
43	2, 4, 27	islln2a 36046	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → ((𝐹 ∨ 𝐺) ∈ (LLines‘𝐾) ↔ 𝐹 ≠ 𝐺))
44	12, 38, 42, 43	syl3anc 1351	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝐹 ∨ 𝐺) ∈ (LLines‘𝐾) ↔ 𝐹 ≠ 𝐺))
45	31, 44	mpbird 249	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝐹 ∨ 𝐺) ∈ (LLines‘𝐾))
46	2, 3, 4, 27, 22	2llnmj 36089	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾) ∧ (𝐹 ∨ 𝐺) ∈ (LLines‘𝐾)) → (((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) ∈ 𝐴 ↔ ((𝑆 ∨ 𝑇) ∨ (𝐹 ∨ 𝐺)) ∈ (LPlanes‘𝐾)))
47	12, 30, 45, 46	syl3anc 1351	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) ∈ 𝐴 ↔ ((𝑆 ∨ 𝑇) ∨ (𝐹 ∨ 𝐺)) ∈ (LPlanes‘𝐾)))
48	26, 47	mpbird 249	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ (𝐹 ∨ 𝐺)) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048   ≠ wne 2961   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970  lecple 16418  joincjn 17402  meetcmee 17403  Atomscatm 35792  HLchlt 35879  LLinesclln 36020  LPlanesclpl 36021  LHypclh 36513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-proset 17386  df-poset 17404  df-plt 17416  df-lub 17432  df-glb 17433  df-join 17434  df-meet 17435  df-p0 17497  df-p1 17498  df-lat 17504  df-clat 17566  df-oposet 35705  df-ol 35707  df-oml 35708  df-covers 35795  df-ats 35796  df-atl 35827  df-cvlat 35851  df-hlat 35880  df-llines 36027  df-lplanes 36028  df-lines 36030  df-psubsp 36032  df-pmap 36033  df-padd 36325  df-lhyp 36517

This theorem is referenced by:  cdleme16e  36811  cdleme16f  36812