lplnnle2at - Mathbox for Norm Megill

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Theorem lplnnle2at 38400

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 1194	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝑃)
2		eqid 2732	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2732	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		eqid 2732	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
5		lplnnle2at.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
6	2, 3, 4, 5	islpln 38389	. . . . 5 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
7	6	adantr 481	. . . 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 496	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10		oveq1 7412	. . . . . . . . 9 ⊢ (𝑄 = 𝑅 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑅))
11	10	breq2d 5159	. . . . . . . 8 ⊢ (𝑄 = 𝑅 → (𝑋 ≤ (𝑄 ∨ 𝑅) ↔ 𝑋 ≤ (𝑅 ∨ 𝑅)))
12	11	notbid 317	. . . . . . 7 ⊢ (𝑄 = 𝑅 → (¬ 𝑋 ≤ (𝑄 ∨ 𝑅) ↔ ¬ 𝑋 ≤ (𝑅 ∨ 𝑅)))
13		simpl1 1191	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝐾 ∈ HL)
14		simpl3l 1228	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦 ∈ (LLines‘𝐾))
15		simpl22 1252	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴)
16		simpl23 1253	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴)
17		simpr 485	. . . . . . . . . 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 38371	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾))
21	13, 15, 16, 17, 20	syl31anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾))
22		eqid 2732	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
23	22, 4	llnnlt 38382	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))
24	13, 14, 21, 23	syl3anc 1371	. . . . . . . 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 38368	. . . . . . . . . . 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 1251	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑋 ∈ 𝑃)
28	2, 5	lplnbase 38393	. . . . . . . . . . 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 1229	. . . . . . . . . 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 38128	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
32	13, 26, 29, 30, 31	syl31anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦(lt‘𝐾)𝑋)
33		hlpos 38224	. . . . . . . . . . 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 38225	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
36	13, 15, 16, 35	syl3anc 1371	. . . . . . . . . 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 18292	. . . . . . . . . 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 1372	. . . . . . . . 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 693	. . . . . . . 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 1136	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
43		simp3l 1201	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾))
44		simp23 1208	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ 𝐴)
45	37, 19, 4	llnnleat 38372	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅 ∈ 𝐴) → ¬ 𝑦 ≤ 𝑅)
46	42, 43, 44, 45	syl3anc 1371	. . . . . . . . 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 1206	. . . . . . . . . . . . 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 1202	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
52	33	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
53	2, 19	atbase 38147	. . . . . . . . . . . . 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 18292	. . . . . . . . . . . 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 1372	. . . . . . . . . . 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 693	. . . . . . . . . 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 18282	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅 ∈ 𝐴) → (𝑦(lt‘𝐾)𝑅 → 𝑦 ≤ 𝑅))
59	42, 43, 44, 58	syl3anc 1371	. . . . . . . . . 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 38227	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) → (𝑅 ∨ 𝑅) = 𝑅)
63	42, 44, 62	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 ∨ 𝑅) = 𝑅)
64	63	breq2d 5159	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑅 ∨ 𝑅) ↔ 𝑋 ≤ 𝑅))
65	61, 64	mtbird 324	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑅 ∨ 𝑅))
66	12, 41, 65	pm2.61ne 3027	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))
67	66	3exp 1119	. . . . 5 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))))
68	67	exp4a 432	. . . 4 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)))))
69	68	imp 407	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))))
70	69	rexlimdv 3153	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 lecple 17200 Posetcpo 18256 ltcplt 18257 joincjn 18260 ⋖ ccvr 38120 Atomscatm 38121 HLchlt 38208 LLinesclln 38350 LPlanesclpl 38351

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-lat 18381 df-clat 18448 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358

This theorem is referenced by: lplnnleat 38401 lplnnlelln 38402 2atnelpln 38403 lvolnle3at 38441

W3C validator