Theorem cdlemblem 40239

Description: Lemma for cdlemb 40240. (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 1311	. . 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 1304	. . . . . . 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 1201	. . . . . . 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 1288	. . . . . . 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 1129	. . . . . 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 1245	. . . . . 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 18297	. . . . . 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 39810	. . . . . . 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 1203	. . . . . . . 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 39735	. . . . . . . 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 39735	. . . . . . . 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 18416	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) ↔ (𝑟 ∨ 𝑢) ≤ 𝑋))
21	11, 16, 18, 4, 20	syl13anc 1375	. . . . . 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 229	. . . . 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 695	. . . 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 1305	. . . . . . . 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 1129	. . . . . . 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 1284	. . . . . . 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 1129	. . . . . 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 1285	. . . . . 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 39842	. . . . . 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 39735	. . . . . . 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 18405	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵) → (𝑟 ∨ 𝑢) ∈ 𝐵)
34	11, 16, 18, 33	syl3anc 1374	. . . . . 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 18410	. . . . . 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 1375	. . . . 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 696	. . . 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 198	. 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 1244	. . 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 1306	. . . . . . . . 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 1283	. . . . . . . . 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 39839	. . . . . . . . 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 1386	. . . . . . . 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 1129	. . . . . . . . . . 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 1129	. . . . . . . . . 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 39841	. . . . . . . . . 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 5089	. . . . . . . . 9 ⊢ ((𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄) → (𝑢 ≤ (𝑃 ∨ 𝑟) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
50	48, 49	syl5ibcom 245	. . . . . . . 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 240	. . . . . . 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 526	. . . . . 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 39735	. . . . . . . . . 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 18405	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
56	11, 32, 54, 55	syl3anc 1374	. . . . . . . 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 18432	. . . . . . . 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 1375	. . . . . . 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 5093	. . . . . . 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 239	. . . . 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 39806	. . . . . . 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 1289	. . . . . . . 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 1310	. . . . . . . 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 39922	. . . . . . . 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 1396	. . . . . . 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 2840	. . . . . 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 39757	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑢 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑢 ≤ 𝑉 ↔ 𝑢 = 𝑉))
74	65, 3, 72, 73	syl3anc 1374	. . . . 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 239	. . . 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 2945	. . 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 511	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  ltcplt 18274  joincjn 18277  meetcmee 18278  1.cp1 18388  Latclat 18397   ⋖ ccvr 39708  Atomscatm 39709  AtLatcal 39710  HLchlt 39796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-p1 18390  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797

This theorem is referenced by:  cdlemb  40240