Theorem lhp2lt 39604

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 1197	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ≤ 𝑊)
2		simp3r 1199	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ≤ 𝑊)
3		simp1l 1194	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝐾 ∈ HL)
4	3	hllatd 38966	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝐾 ∈ Lat)
5		simp2l 1196	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
6		eqid 2725	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		lhp2lt.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 38891	. . . . 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 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → 𝑄 ∈ 𝐴)
11	6, 7	atbase 38891	. . . . 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 1195	. . . . 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 39601	. . . . 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 18445	. . . 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 1369	. . 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 710	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) ≤ 𝑊)
22	18, 17, 7	3dim2 39071	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
23	3, 5, 10, 22	syl3anc 1368	. . 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 1281	. . . . . . . 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 38964	. . . . . . . 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 38966	. . . . . . . 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 1283	. . . . . . . . . 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 1285	. . . . . . . . . 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 38969	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
31	24, 28, 29, 30	syl3anc 1368	. . . . . . . . 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 1196	. . . . . . . . . 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 38891	. . . . . . . . . 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 18434	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾))
36	27, 31, 34, 35	syl3anc 1368	. . . . . . . 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 1197	. . . . . . . . 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 38891	. . . . . . . . 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 18434	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑟) ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾)) → (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾))
41	27, 36, 39, 40	syl3anc 1368	. . . . . . 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 2725	. . . . . . . 8 ⊢ (1.‘𝐾) = (1.‘𝐾)
43		eqid 2725	. . . . . . . 8 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
44	6, 42, 43	ncvr1 38874	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠) ∈ (Base‘𝐾)) → ¬ (1.‘𝐾)( ⋖ ‘𝐾)(((𝑃 ∨ 𝑄) ∨ 𝑟) ∨ 𝑠))
45	26, 41, 44	syl2anc 582	. . . . . 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 2725	. . . . . . . . . . . 12 ⊢ (lub‘𝐾) = (lub‘𝐾)
47		simpl1l 1221	. . . . . . . . . . . 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 38966	. . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . 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 1225	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 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 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)) → 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 1368	. . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . 15 ⊢ (glb‘𝐾) = (glb‘𝐾)
57	6, 46, 56	op01dm 38785	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ OP → ((Base‘𝐾) ∈ dom (lub‘𝐾) ∧ (Base‘𝐾) ∈ dom (glb‘𝐾)))
58	57	simpld 493	. . . . . . . . . . . . 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 18425	. . . . . . . . . . 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 38968	. . . . . . . . . . . . 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 38787	. . . . . . . . . . . . 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 1229	. . . . . . . . . . . . 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 39013	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑟 ∈ 𝐴) → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑄)( ⋖ ‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑟)))
67	47, 51, 52, 66	syl3anc 1368	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 1193	. . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . 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 39602	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾))
72	47, 70, 71	syl2anc 582	. . . . . . . . . . . . 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 5171	. . . . . . . . . . . 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 38885	. . . . . . . . . . . 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 1385	. . . . . . . . . . 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 231	. . . . . . . . . 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 1230	. . . . . . . . . . 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 1228	. . . . . . . . . . . 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 39013	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 231	. . . . . . . . . 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 5173	. . . . . . . . 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 1351	. . . . . . . 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 1108	. . . . . . 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 2943	. . . . . 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 1116	. . . 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 3200	. . 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 1368	. . 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 18327	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ 𝐻) → ((𝑃 ∨ 𝑄) < 𝑊 ↔ ((𝑃 ∨ 𝑄) ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) ≠ 𝑊)))
93	3, 90, 13, 92	syl3anc 1368	. 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 711	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (𝑃 ∨ 𝑄) < 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∃wrex 3059   class class class wbr 5149  dom cdm 5678  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  Posetcpo 18302  ltcplt 18303  lubclub 18304  glbcglb 18305  joincjn 18306  1.cp1 18419  Latclat 18426  OPcops 38774   ⋖ ccvr 38864  Atomscatm 38865  HLchlt 38952  LHypclh 39587

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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-p1 18421  df-lat 18427  df-clat 18494  df-oposet 38778  df-ol 38780  df-oml 38781  df-covers 38868  df-ats 38869  df-atl 38900  df-cvlat 38924  df-hlat 38953  df-lhyp 39591

This theorem is referenced by:  lhpexle3lem  39614