Theorem cdlemblem 37803

Description: Lemma for cdlemb 37804. (Contributed by NM, 8-May-2012.)

Hypotheses

Ref	Expression
cdlemb.b	⊢ 𝐵 = (Base‘𝐾)
cdlemb.l	⊢ ≤ = (le‘𝐾)
cdlemb.j	⊢ ∨ = (join‘𝐾)
cdlemb.u	⊢ 1 = (1.‘𝐾)
cdlemb.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
cdlemb.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemblem.s	⊢ < = (lt‘𝐾)
cdlemblem.m	⊢ ∧ = (meet‘𝐾)
cdlemblem.v	⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑋)

Assertion

Ref	Expression
cdlemblem	⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Proof of Theorem cdlemblem

Step	Hyp	Ref	Expression
1		simp132 1308	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ¬ 𝑃 ≤ 𝑋)
2		simp111 1301	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝐾 ∈ HL)
3		simp2l 1198	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ∈ 𝐴)
4		simp12l 1285	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑋 ∈ 𝐵)
5	2, 3, 4	3jca 1127	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝐾 ∈ HL ∧ 𝑢 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵))
6		simp2rr 1242	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 < 𝑋)
7		cdlemb.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
8		cdlemblem.s	. . . . . . 7 ⊢ < = (lt‘𝐾)
9	7, 8	pltle 18049	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑢 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑢 < 𝑋 → 𝑢 ≤ 𝑋))
10	5, 6, 9	sylc 65	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ≤ 𝑋)
11	2	hllatd 37374	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝐾 ∈ Lat)
12		simp3l 1200	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ∈ 𝐴)
13		cdlemb.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
14		cdlemb.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
15	13, 14	atbase 37299	. . . . . . . 8 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵)
16	12, 15	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ∈ 𝐵)
17	13, 14	atbase 37299	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵)
18	3, 17	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ∈ 𝐵)
19		cdlemb.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
20	13, 7, 19	latjle12 18166	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) ↔ (𝑟 ∨ 𝑢) ≤ 𝑋))
21	11, 16, 18, 4, 20	syl13anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) ↔ (𝑟 ∨ 𝑢) ≤ 𝑋))
22	21	biimpd 228	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) → (𝑟 ∨ 𝑢) ≤ 𝑋))
23	10, 22	mpan2d 691	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ 𝑋 → (𝑟 ∨ 𝑢) ≤ 𝑋))
24		simp112 1302	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ∈ 𝐴)
25	12, 24, 3	3jca 1127	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴))
26		simp3r2 1281	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ≠ 𝑢)
27	2, 25, 26	3jca 1127	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑟 ≠ 𝑢))
28		simp3r3 1282	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ≤ (𝑃 ∨ 𝑢))
29	7, 19, 14	hlatexch2 37406	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑟 ≠ 𝑢) → (𝑟 ≤ (𝑃 ∨ 𝑢) → 𝑃 ≤ (𝑟 ∨ 𝑢)))
30	27, 28, 29	sylc 65	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ≤ (𝑟 ∨ 𝑢))
31	13, 14	atbase 37299	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
32	24, 31	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ∈ 𝐵)
33	13, 19	latjcl 18155	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵) → (𝑟 ∨ 𝑢) ∈ 𝐵)
34	11, 16, 18, 33	syl3anc 1370	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ∨ 𝑢) ∈ 𝐵)
35	13, 7	lattr 18160	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑟 ∨ 𝑢) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ (𝑟 ∨ 𝑢) ∧ (𝑟 ∨ 𝑢) ≤ 𝑋) → 𝑃 ≤ 𝑋))
36	11, 32, 34, 4, 35	syl13anc 1371	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑃 ≤ (𝑟 ∨ 𝑢) ∧ (𝑟 ∨ 𝑢) ≤ 𝑋) → 𝑃 ≤ 𝑋))
37	30, 36	mpand 692	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑟 ∨ 𝑢) ≤ 𝑋 → 𝑃 ≤ 𝑋))
38	23, 37	syld 47	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ 𝑋 → 𝑃 ≤ 𝑋))
39	1, 38	mtod 197	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ¬ 𝑟 ≤ 𝑋)
40		simp2rl 1241	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ≠ 𝑉)
41		simp113 1303	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑄 ∈ 𝐴)
42		simp3r1 1280	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ≠ 𝑃)
43	7, 19, 14	hlatexchb1 37403	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ≠ 𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄)))
44	2, 12, 41, 24, 42, 43	syl131anc 1382	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄)))
45	12, 3, 24	3jca 1127	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴))
46	2, 45, 42	3jca 1127	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ≠ 𝑃))
47	7, 19, 14	hlatexch1 37405	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ≠ 𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑢) → 𝑢 ≤ (𝑃 ∨ 𝑟)))
48	46, 28, 47	sylc 65	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ≤ (𝑃 ∨ 𝑟))
49		breq2 5083	. . . . . . . . 9 ⊢ ((𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄) → (𝑢 ≤ (𝑃 ∨ 𝑟) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
50	48, 49	syl5ibcom 244	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄) → 𝑢 ≤ (𝑃 ∨ 𝑄)))
51	44, 50	sylbid 239	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑢 ≤ (𝑃 ∨ 𝑄)))
52	51, 10	jctird 527	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋)))
53	13, 14	atbase 37299	. . . . . . . . . 10 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
54	41, 53	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑄 ∈ 𝐵)
55	13, 19	latjcl 18155	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
56	11, 32, 54, 55	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑃 ∨ 𝑄) ∈ 𝐵)
57		cdlemblem.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
58	13, 7, 57	latlem12 18182	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑢 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋) ↔ 𝑢 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
59	11, 18, 56, 4, 58	syl13anc 1371	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋) ↔ 𝑢 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
60		cdlemblem.v	. . . . . . . 8 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑋)
61	60	breq2i 5087	. . . . . . 7 ⊢ (𝑢 ≤ 𝑉 ↔ 𝑢 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋))
62	59, 61	bitr4di 289	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋) ↔ 𝑢 ≤ 𝑉))
63	52, 62	sylibd 238	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑢 ≤ 𝑉))
64		hlatl 37370	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
65	2, 64	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝐾 ∈ AtLat)
66		simp12r 1286	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ≠ 𝑄)
67		simp131 1307	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑋𝐶 1 )
68		cdlemb.u	. . . . . . . . 9 ⊢ 1 = (1.‘𝐾)
69		cdlemb.c	. . . . . . . . 9 ⊢ 𝐶 = ( ⋖ ‘𝐾)
70	13, 7, 19, 57, 68, 69, 14	1cvrat 37486	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
71	2, 24, 41, 4, 66, 67, 1, 70	syl133anc 1392	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
72	60, 71	eqeltrid 2845	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑉 ∈ 𝐴)
73	7, 14	atcmp 37321	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑢 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑢 ≤ 𝑉 ↔ 𝑢 = 𝑉))
74	65, 3, 72, 73	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑢 ≤ 𝑉 ↔ 𝑢 = 𝑉))
75	63, 74	sylibd 238	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑢 = 𝑉))
76	75	necon3ad 2958	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑢 ≠ 𝑉 → ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
77	40, 76	mpd 15	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))
78	39, 77	jca 512	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110   ≠ wne 2945   class class class wbr 5079  ‘cfv 6432  (class class class)co 7271  Basecbs 16910  lecple 16967  ltcplt 18024  joincjn 18027  meetcmee 18028  1.cp1 18140  Latclat 18147   ⋖ ccvr 37272  Atomscatm 37273  AtLatcal 37274  HLchlt 37360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-proset 18011  df-poset 18029  df-plt 18046  df-lub 18062  df-glb 18063  df-join 18064  df-meet 18065  df-p0 18141  df-p1 18142  df-lat 18148  df-clat 18215  df-oposet 37186  df-ol 37188  df-oml 37189  df-covers 37276  df-ats 37277  df-atl 37308  df-cvlat 37332  df-hlat 37361

This theorem is referenced by:  cdlemb  37804