lplnnle2at - Mathbox for Norm Megill

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Theorem lplnnle2at 38868

Description: A lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)

**Hypotheses**
Ref	Expression
lplnnle2at.l	⊢ ≤ = (le‘𝐾)
lplnnle2at.j	⊢ ∨ = (join‘𝐾)
lplnnle2at.a	⊢ 𝐴 = (Atoms‘𝐾)
lplnnle2at.p	⊢ 𝑃 = (LPlanes‘𝐾)

**Assertion**
Ref	Expression
lplnnle2at	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))

Proof of Theorem lplnnle2at Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1191	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝑃)
2		eqid 2724	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2724	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		eqid 2724	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
5		lplnnle2at.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
6	2, 3, 4, 5	islpln 38857	. . . . 5 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
7	6	adantr 480	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
8	1, 7	mpbid 231	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
9	8	simprd 495	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10		oveq1 7408	. . . . . . . . 9 ⊢ (𝑄 = 𝑅 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑅))
11	10	breq2d 5150	. . . . . . . 8 ⊢ (𝑄 = 𝑅 → (𝑋 ≤ (𝑄 ∨ 𝑅) ↔ 𝑋 ≤ (𝑅 ∨ 𝑅)))
12	11	notbid 318	. . . . . . 7 ⊢ (𝑄 = 𝑅 → (¬ 𝑋 ≤ (𝑄 ∨ 𝑅) ↔ ¬ 𝑋 ≤ (𝑅 ∨ 𝑅)))
13		simpl1 1188	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝐾 ∈ HL)
14		simpl3l 1225	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦 ∈ (LLines‘𝐾))
15		simpl22 1249	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴)
16		simpl23 1250	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴)
17		simpr 484	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ≠ 𝑅)
18		lplnnle2at.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
19		lplnnle2at.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Atoms‘𝐾)
20	18, 19, 4	llni2 38839	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾))
21	13, 15, 16, 17, 20	syl31anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾))
22		eqid 2724	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
23	22, 4	llnnlt 38850	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))
24	13, 14, 21, 23	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → ¬ 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))
25	2, 4	llnbase 38836	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (LLines‘𝐾) → 𝑦 ∈ (Base‘𝐾))
26	14, 25	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦 ∈ (Base‘𝐾))
27		simpl21 1248	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑋 ∈ 𝑃)
28	2, 5	lplnbase 38861	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑋 ∈ (Base‘𝐾))
30		simpl3r 1226	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦( ⋖ ‘𝐾)𝑋)
31	2, 22, 3	cvrlt 38596	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
32	13, 26, 29, 30, 31	syl31anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦(lt‘𝐾)𝑋)
33		hlpos 38692	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
34	13, 33	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝐾 ∈ Poset)
35	2, 18, 19	hlatjcl 38693	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
36	13, 15, 16, 35	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
37		lplnnle2at.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
38	2, 37, 22	pltletr 18297	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)))
39	34, 26, 29, 36, 38	syl13anc 1369	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)))
40	32, 39	mpand 692	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑋 ≤ (𝑄 ∨ 𝑅) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)))
41	24, 40	mtod 197	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))
42		simp1 1133	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
43		simp3l 1198	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾))
44		simp23 1205	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ 𝐴)
45	37, 19, 4	llnnleat 38840	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅 ∈ 𝐴) → ¬ 𝑦 ≤ 𝑅)
46	42, 43, 44, 45	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 ≤ 𝑅)
47	43, 25	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (Base‘𝐾))
48		simp21 1203	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ 𝑃)
49	48, 28	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
50		simp3r 1199	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
51	42, 47, 49, 50, 31	syl31anc 1370	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
52	33	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
53	2, 19	atbase 38615	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
54	44, 53	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
55	2, 37, 22	pltletr 18297	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑅) → 𝑦(lt‘𝐾)𝑅))
56	52, 47, 49, 54, 55	syl13anc 1369	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑅) → 𝑦(lt‘𝐾)𝑅))
57	51, 56	mpand 692	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ 𝑅 → 𝑦(lt‘𝐾)𝑅))
58	37, 22	pltle 18287	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅 ∈ 𝐴) → (𝑦(lt‘𝐾)𝑅 → 𝑦 ≤ 𝑅))
59	42, 43, 44, 58	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)𝑅 → 𝑦 ≤ 𝑅))
60	57, 59	syld 47	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ 𝑅 → 𝑦 ≤ 𝑅))
61	46, 60	mtod 197	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ 𝑅)
62	18, 19	hlatjidm 38695	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) → (𝑅 ∨ 𝑅) = 𝑅)
63	42, 44, 62	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 ∨ 𝑅) = 𝑅)
64	63	breq2d 5150	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑅 ∨ 𝑅) ↔ 𝑋 ≤ 𝑅))
65	61, 64	mtbird 325	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑅 ∨ 𝑅))
66	12, 41, 65	pm2.61ne 3019	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))
67	66	3exp 1116	. . . . 5 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))))
68	67	exp4a 431	. . . 4 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)))))
69	68	imp 406	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))))
70	69	rexlimdv 3145	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)))
71	9, 70	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 lecple 17202 Posetcpo 18261 ltcplt 18262 joincjn 18265 ⋖ ccvr 38588 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: lplnnleat 38869 lplnnlelln 38870 2atnelpln 38871 lvolnle3at 38909

W3C validator