Theorem lhp2lt 40833

Description: The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)

Hypotheses

Ref	Expression
lhp2lt.l	⊢ ≤ = (le‘𝐾)
lhp2lt.s	⊢ < = (lt‘𝐾)
lhp2lt.j	⊢ ∨ = (join‘𝐾)
lhp2lt.a	⊢ 𝐴 = (Atoms‘𝐾)
lhp2lt.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhp2lt	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) < 𝑊)

Proof of Theorem lhp2lt

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2r 1219	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ≤ 𝑊)
2		simp3r 1221	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ≤ 𝑊)
3		simp1l 1216	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝐾 ∈ HL)
4	3	hllatd 40196	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝐾 ∈ Lat)
5		simp2l 1218	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
6		eqid 2765	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		lhp2lt.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 40121	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
9	5, 8	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ∈ (Base‘𝐾))
10		simp3l 1220	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ∈ 𝐴)
11	6, 7	atbase 40121	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
12	10, 11	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ∈ (Base‘𝐾))
13		simp1r 1217	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
14		lhp2lt.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
15	6, 14	lhpbase 40830	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
16	13, 15	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾))
17		lhp2lt.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
18		lhp2lt.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
19	6, 17, 18	latjle12 18528	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
20	4, 9, 12, 16, 19	syl13anc 1399	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊))
21	1, 2, 20	mpbi2and 725	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ≤ 𝑊)
22	18, 17, 7	3dim2 40300	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
23	3, 5, 10, 22	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
24		simp11l 1303	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝐾 ∈ HL)
25		hlop 40194	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
26	24, 25	syl 18	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝐾 ∈ OP)
27	24	hllatd 40196	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝐾 ∈ Lat)
28		simp12l 1305	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝑃 ∈ 𝐴)
29		simp13l 1307	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝑄 ∈ 𝐴)
30	6, 18, 7	hlatjcl 40199	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
31	24, 28, 29, 30	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
32		simp2l 1218	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝑟 ∈ 𝐴)
33	6, 7	atbase 40121	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
34	32, 33	syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝑟 ∈ (Base‘𝐾))
35	6, 18	latjcl 18517	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾))
36	27, 31, 34, 35	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾))
37		simp2r 1219	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝑠 ∈ 𝐴)
38	6, 7	atbase 40121	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
39	37, 38	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → 𝑠 ∈ (Base‘𝐾))
40	6, 18	latjcl 18517	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾)) → (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾))
41	27, 36, 39, 40	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾))
42		eqid 2765	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
43		eqid 2765	. . . . . . . 8 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
44	6, 42, 43	ncvr1 40104	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾)) → ¬ (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠))
45	26, 41, 44	syl2anc 596	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → ¬ (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠))
46		eqid 2765	. . . . . . . . . . . 12 ⊢ (lub‘𝐾) = (lub‘𝐾)
47		simpl1l 1243	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝐾 ∈ HL)
48	47	hllatd 40196	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝐾 ∈ Lat)
49		simpl2l 1245	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑃 ∈ 𝐴)
50		simpl3l 1247	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑄 ∈ 𝐴)
51	47, 49, 50, 30	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
52		simpr1l 1249	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑟 ∈ 𝐴)
53	52, 33	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑟 ∈ (Base‘𝐾))
54	48, 51, 53, 35	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾))
55	47, 25	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝐾 ∈ OP)
56		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (glb‘𝐾) = (glb‘𝐾)
57	6, 46, 56	op01dm 40015	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ OP → ((Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)))
58	57	simpld 500	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OP → (Base‘𝐾) ∈ dom (lub‘𝐾))
59	55, 58	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (Base‘𝐾) ∈ dom (lub‘𝐾))
60	6, 46, 17, 42, 47, 54, 59	ple1 18506	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → ((𝑃 ∨ 𝑄) ∨ 𝑟) ≤ (1.‘𝐾))
61		hlpos 40198	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
62	47, 61	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝐾 ∈ Poset)
63	6, 42	op1cl 40017	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OP → (1.‘𝐾) ∈ (Base‘𝐾))
64	55, 63	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (1.‘𝐾) ∈ (Base‘𝐾))
65		simpr2l 1251	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))
66	6, 17, 18, 43, 7	cvr1 40242	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑟 ∈ 𝐴) → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑟)))
67	47, 51, 52, 66	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑟)))
68	65, 67	mpbid 235	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑟))
69		simpr3 1215	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (𝑃 ∨ 𝑄) = 𝑊)
70		simpl1r 1244	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑊 ∈ 𝐻)
71	42, 43, 14	lhp1cvr 40831	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾))
72	47, 70, 71	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾))
73	69, 72	eqbrtrd 5135	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)(1.‘𝐾))
74	6, 17, 43	cvrcmp 40115	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Poset ∧ (((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾) ∧ (1.‘𝐾) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) ∧ ((𝑃 ∨ 𝑄)( ⋖ ‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑟) ∧ (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)(1.‘𝐾))) → (((𝑃 ∨ 𝑄) ∨ 𝑟) ≤ (1.‘𝐾) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑟) = (1.‘𝐾)))
75	62, 54, 64, 51, 68, 73, 74	syl132anc 1415	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (((𝑃 ∨ 𝑄) ∨ 𝑟) ≤ (1.‘𝐾) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑟) = (1.‘𝐾)))
76	60, 75	mpbid 235	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → ((𝑃 ∨ 𝑄) ∨ 𝑟) = (1.‘𝐾))
77		simpr2r 1252	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))
78		simpr1r 1250	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → 𝑠 ∈ 𝐴)
79	6, 17, 18, 43, 7	cvr1 40242	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾) ∧ 𝑠 ∈ 𝐴) → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑟)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠)))
80	47, 54, 78, 79	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑟)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠)))
81	77, 80	mpbid 235	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → ((𝑃 ∨ 𝑄) ∨ 𝑟)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠))
82	76, 81	eqbrtrrd 5137	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ∧ (𝑃 ∨ 𝑄) = 𝑊)) → (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠))
83	82	3exp2 1373	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → ((¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) → ((𝑃 ∨ 𝑄) = 𝑊 → (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠)))))
84	83	3imp 1128	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → ((𝑃 ∨ 𝑄) = 𝑊 → (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠)))
85	84	necon3bd 2974	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → (¬ (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) → (𝑃 ∨ 𝑄) ≠ 𝑊))
86	45, 85	mpd 16	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))) → (𝑃 ∨ 𝑄) ≠ 𝑊)
87	86	3exp 1137	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → ((¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) → (𝑃 ∨ 𝑄) ≠ 𝑊)))
88	87	rexlimdvv 3223	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) → (𝑃 ∨ 𝑄) ≠ 𝑊))
89	23, 88	mpd 16	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ≠ 𝑊)
90	3, 5, 10, 30	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
91		lhp2lt.s	. . . 4 ⊢ < = (lt‘𝐾)
92	17, 91	pltval 18408	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ 𝐻) → ((𝑃 ∨ 𝑄) < 𝑊 ↔ ((𝑃 ∨ 𝑄) ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) ≠ 𝑊)))
93	3, 90, 13, 92	syl3anc 1398	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ((𝑃 ∨ 𝑄) < 𝑊 ↔ ((𝑃 ∨ 𝑄) ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) ≠ 𝑊)))
94	21, 89, 93	mpbir2and 726	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) < 𝑊)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   class class class wbr 5111  dom cdm 5663  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  lecple 17339  Posetcpo 18385  ltcplt 18386  lubclub 18387  glbcglb 18388  joincjn 18389  1.cp1 18500  Latclat 18509  OPcops 40004   ⋖ ccvr 40094  Atomscatm 40095  HLchlt 40182  LHypclh 40816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-proset 18372  df-poset 18391  df-plt 18406  df-lub 18422  df-glb 18423  df-join 18424  df-meet 18425  df-p0 18501  df-p1 18502  df-lat 18510  df-clat 18577  df-oposet 40008  df-ol 40010  df-oml 40011  df-covers 40098  df-ats 40099  df-atl 40130  df-cvlat 40154  df-hlat 40183  df-lhyp 40820

This theorem is used by:  lhpexle3lem  40843