Theorem lhp2lt 39980

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 1201	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ≤ 𝑊)
2		simp3r 1203	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ≤ 𝑊)
3		simp1l 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝐾 ∈ HL)
4	3	hllatd 39343	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝐾 ∈ Lat)
5		simp2l 1200	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
6		eqid 2729	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		lhp2lt.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 39268	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
9	5, 8	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ∈ (Base‘𝐾))
10		simp3l 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ∈ 𝐴)
11	6, 7	atbase 39268	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
12	10, 11	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ∈ (Base‘𝐾))
13		simp1r 1199	. . . . 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 39977	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
16	13, 15	syl 17	. . . 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 18356	. . . 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 1374	. . 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 712	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ≤ 𝑊)
22	18, 17, 7	3dim2 39447	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
23	3, 5, 10, 22	syl3anc 1373	. . 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 1285	. . . . . . . 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 39341	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
26	24, 25	syl 17	. . . . . . 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 39343	. . . . . . . 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 1287	. . . . . . . . . 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 1289	. . . . . . . . . 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 39346	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
31	24, 28, 29, 30	syl3anc 1373	. . . . . . . . 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 1200	. . . . . . . . . 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 39268	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
34	32, 33	syl 17	. . . . . . . . 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 18345	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾))
36	27, 31, 34, 35	syl3anc 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) ∨ 𝑟) ∈ (Base‘𝐾))
37		simp2r 1201	. . . . . . . . 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 39268	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
39	37, 38	syl 17	. . . . . . . 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 18345	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾)) → (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾))
41	27, 36, 39, 40	syl3anc 1373	. . . . . . 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 2729	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
43		eqid 2729	. . . . . . . 8 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
44	6, 42, 43	ncvr1 39251	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾)) → ¬ (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠))
45	26, 41, 44	syl2anc 584	. . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (lub‘𝐾) = (lub‘𝐾)
47		simpl1l 1225	. . . . . . . . . . . 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 39343	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 15 ⊢ (glb‘𝐾) = (glb‘𝐾)
57	6, 46, 56	op01dm 39162	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ OP → ((Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)))
58	57	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OP → (Base‘𝐾) ∈ dom (lub‘𝐾))
59	55, 58	syl 17	. . . . . . . . . . . 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 18334	. . . . . . . . . . 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 39345	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
62	47, 61	syl 17	. . . . . . . . . . . 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 39164	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OP → (1.‘𝐾) ∈ (Base‘𝐾))
64	55, 63	syl 17	. . . . . . . . . . . 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 1233	. . . . . . . . . . . . 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 39389	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑟 ∈ 𝐴) → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑟)))
67	47, 51, 52, 66	syl3anc 1373	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 1197	. . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . 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 39978	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾))
72	47, 70, 71	syl2anc 584	. . . . . . . . . . . . 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 5114	. . . . . . . . . . . 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 39262	. . . . . . . . . . . 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 1390	. . . . . . . . . . 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 232	. . . . . . . . . 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 1234	. . . . . . . . . . 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 1232	. . . . . . . . . . . 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 39389	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 232	. . . . . . . . . 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 5116	. . . . . . . . 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 1355	. . . . . . . 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 1110	. . . . . . 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 2939	. . . . . 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 15	. . . . 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 1119	. . . 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 3185	. . 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 15	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ≠ 𝑊)
90	3, 5, 10, 30	syl3anc 1373	. . 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 18236	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ 𝐻) → ((𝑃 ∨ 𝑄) < 𝑊 ↔ ((𝑃 ∨ 𝑄) ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) ≠ 𝑊)))
93	3, 90, 13, 92	syl3anc 1373	. 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 713	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) < 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5092  dom cdm 5619  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  Posetcpo 18213  ltcplt 18214  lubclub 18215  glbcglb 18216  joincjn 18217  1.cp1 18328  Latclat 18337  OPcops 39151   ⋖ ccvr 39241  Atomscatm 39242  HLchlt 39329  LHypclh 39963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39155  df-ol 39157  df-oml 39158  df-covers 39245  df-ats 39246  df-atl 39277  df-cvlat 39301  df-hlat 39330  df-lhyp 39967

This theorem is referenced by:  lhpexle3lem  39990