Theorem lvolnle3at 36270

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

Hypotheses

Ref	Expression
lvolnle3at.l	⊢ ≤ = (le‘𝐾)
lvolnle3at.j	⊢ ∨ = (join‘𝐾)
lvolnle3at.a	⊢ 𝐴 = (Atoms‘𝐾)
lvolnle3at.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lvolnle3at	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Proof of Theorem lvolnle3at

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 765	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝑉)
2		eqid 2797	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2797	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4		eqid 2797	. . . . . 6 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾)
5		lvolnle3at.v	. . . . . 6 ⊢ 𝑉 = (LVols‘𝐾)
6	2, 3, 4, 5	islvol 36261	. . . . 5 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
7	6	ad2antrr 722	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
8	1, 7	mpbid 233	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
9	8	simprd 496	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10		oveq1 7030	. . . . . . . . 9 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
11	10	oveq1d 7038	. . . . . . . 8 ⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅))
12	11	breq2d 4980	. . . . . . 7 ⊢ (𝑃 = 𝑄 → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
13	12	notbid 319	. . . . . 6 ⊢ (𝑃 = 𝑄 → (¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
14		simp1l 1190	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
15		simp3l 1194	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LPlanes‘𝐾))
16		simp21 1199	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑃 ∈ 𝐴)
17		simp22 1200	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑄 ∈ 𝐴)
18		lvolnle3at.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
19		lvolnle3at.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
20		lvolnle3at.a	. . . . . . . . . . . . 13 ⊢ 𝐴 = (Atoms‘𝐾)
21	18, 19, 20, 4	lplnnle2at 36229	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ¬ 𝑦 ≤ (𝑃 ∨ 𝑄))
22	14, 15, 16, 17, 21	syl13anc 1365	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 ≤ (𝑃 ∨ 𝑄))
23	2, 4	lplnbase 36222	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (LPlanes‘𝐾) → 𝑦 ∈ (Base‘𝐾))
24	15, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (Base‘𝐾))
25		simp1r 1191	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ 𝑉)
26	2, 5	lvolbase 36266	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾))
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
28		simp3r 1195	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
29		eqid 2797	. . . . . . . . . . . . . . 15 ⊢ (lt‘𝐾) = (lt‘𝐾)
30	2, 29, 3	cvrlt 35958	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
31	14, 24, 27, 28, 30	syl31anc 1366	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
32		hlpos 36054	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
33	14, 32	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
34	2, 19, 20	hlatjcl 36055	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
35	14, 16, 17, 34	syl3anc 1364	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
36	2, 18, 29	pltletr 17414	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑃 ∨ 𝑄)) → 𝑦(lt‘𝐾)(𝑃 ∨ 𝑄)))
37	33, 24, 27, 35, 36	syl13anc 1365	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑃 ∨ 𝑄)) → 𝑦(lt‘𝐾)(𝑃 ∨ 𝑄)))
38	31, 37	mpand 691	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑃 ∨ 𝑄) → 𝑦(lt‘𝐾)(𝑃 ∨ 𝑄)))
39	18, 29	pltle 17404	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑃 ∨ 𝑄) → 𝑦 ≤ (𝑃 ∨ 𝑄)))
40	14, 15, 35, 39	syl3anc 1364	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑃 ∨ 𝑄) → 𝑦 ≤ (𝑃 ∨ 𝑄)))
41	38, 40	syld 47	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑃 ∨ 𝑄) → 𝑦 ≤ (𝑃 ∨ 𝑄)))
42	22, 41	mtod 199	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑃 ∨ 𝑄))
43	42	adantr 481	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑋 ≤ (𝑃 ∨ 𝑄))
44		simprr 769	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
45	14	hllatd 36052	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Lat)
46		simp23 1201	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ 𝐴)
47	2, 20	atbase 35977	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
49	2, 18, 19	latleeqj2 17507	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄)))
50	45, 48, 35, 49	syl3anc 1364	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄)))
51	50	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄)))
52	44, 51	mpbid 233	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄))
53	52	breq2d 4980	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑋 ≤ (𝑃 ∨ 𝑄)))
54	43, 53	mtbird 326	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
55	54	anassrs 468	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
56		simpl1l 1217	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
57		simpl3l 1221	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑦 ∈ (LPlanes‘𝐾))
58		simpl2 1185	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
59		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)))
60	18, 19, 20, 4	lplni2 36225	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (LPlanes‘𝐾))
61	56, 58, 59, 60	syl3anc 1364	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (LPlanes‘𝐾))
62	29, 4	lplnnlt 36253	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (LPlanes‘𝐾)) → ¬ 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅))
63	56, 57, 61, 62	syl3anc 1364	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅))
64	2, 19	latjcl 17494	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾))
65	45, 35, 48, 64	syl3anc 1364	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾))
66	2, 18, 29	pltletr 17414	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅)))
67	33, 24, 27, 65, 66	syl13anc 1365	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅)))
68	31, 67	mpand 691	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅)))
69	68	adantr 481	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅)))
70	63, 69	mtod 199	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
71	70	anassrs 468	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
72	55, 71	pm2.61dan 809	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
73		eqid 2797	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
74	73, 19, 20, 4	lplnnle2at 36229	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑦(le‘𝐾)(𝑄 ∨ 𝑅))
75	14, 15, 17, 46, 74	syl13anc 1365	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦(le‘𝐾)(𝑄 ∨ 𝑅))
76	2, 19, 20	hlatjcl 36055	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
77	14, 17, 46, 76	syl3anc 1364	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
78	2, 18, 29	pltletr 17414	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)))
79	33, 24, 27, 77, 78	syl13anc 1365	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)))
80	31, 79	mpand 691	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑄 ∨ 𝑅) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)))
81	73, 29	pltle 17404	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑄 ∨ 𝑅) → 𝑦(le‘𝐾)(𝑄 ∨ 𝑅)))
82	14, 15, 77, 81	syl3anc 1364	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑄 ∨ 𝑅) → 𝑦(le‘𝐾)(𝑄 ∨ 𝑅)))
83	80, 82	syld 47	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑄 ∨ 𝑅) → 𝑦(le‘𝐾)(𝑄 ∨ 𝑅)))
84	75, 83	mtod 199	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))
85	19, 20	hlatjidm 36057	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
86	14, 17, 85	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 ∨ 𝑄) = 𝑄)
87	86	oveq1d 7038	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑄 ∨ 𝑄) ∨ 𝑅) = (𝑄 ∨ 𝑅))
88	87	breq2d 4980	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅) ↔ 𝑋 ≤ (𝑄 ∨ 𝑅)))
89	84, 88	mtbird 326	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅))
90	13, 72, 89	pm2.61ne 3072	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
91	90	3expia 1114	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
92	91	expd 416	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑦 ∈ (LPlanes‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))))
93	92	rexlimdv 3248	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
94	9, 93	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∃wrex 3108   class class class wbr 4968  ‘cfv 6232  (class class class)co 7023  Basecbs 16316  lecple 16405  Posetcpo 17383  ltcplt 17384  joincjn 17387  Latclat 17488   ⋖ ccvr 35950  Atomscatm 35951  HLchlt 36038  LPlanesclpl 36180  LVolsclvol 36181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-lat 17489  df-clat 17551  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039  df-llines 36186  df-lplanes 36187  df-lvols 36188

This theorem is referenced by:  lvolnleat  36271  lvolnlelln  36272  lvolnlelpln  36273  3atnelvolN  36274  4atlem3  36284  dalem39  36399