Theorem exatleN 38742

Description: A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
atomle.b	⊢ 𝐵 = (Base‘𝐾)
atomle.l	⊢ ≤ = (le‘𝐾)
atomle.j	⊢ ∨ = (join‘𝐾)
atomle.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
exatleN	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃))

Proof of Theorem exatleN

Step	Hyp	Ref	Expression
1		simpl32 1254	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃) → ¬ 𝑄 ≤ 𝑋)
2		atomle.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3		atomle.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
4		simp11l 1283	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝐾 ∈ HL)
5	4	hllatd 38701	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝐾 ∈ Lat)
6		simp122 1305	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑄 ∈ 𝐴)
7		atomle.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8	2, 7	atbase 38626	. . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
9	6, 8	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑄 ∈ 𝐵)
10		simp121 1304	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑃 ∈ 𝐴)
11	2, 7	atbase 38626	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
12	10, 11	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑃 ∈ 𝐵)
13		simp123 1306	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑅 ∈ 𝐴)
14	2, 7	atbase 38626	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑅 ∈ 𝐵)
16		atomle.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
17	2, 16	latjcl 18402	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑅 ∈ 𝐵) → (𝑃 ∨ 𝑅) ∈ 𝐵)
18	5, 12, 15, 17	syl3anc 1370	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → (𝑃 ∨ 𝑅) ∈ 𝐵)
19		simp11r 1284	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑋 ∈ 𝐵)
20	13, 6, 10	3jca 1127	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → (𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴))
21		simp2 1136	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑅 ≠ 𝑃)
22	4, 20, 21	3jca 1127	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → (𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 ≠ 𝑃))
23		simp133 1309	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ (𝑃 ∨ 𝑄))
24	3, 16, 7	hlatexch1 38733	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 ≠ 𝑃) → (𝑅 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑅)))
25	22, 23, 24	sylc 65	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑄 ≤ (𝑃 ∨ 𝑅))
26		simp131 1307	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑃 ≤ 𝑋)
27		simp3 1137	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑅 ≤ 𝑋)
28	2, 3, 16	latjle12 18413	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑅 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑅 ≤ 𝑋) ↔ (𝑃 ∨ 𝑅) ≤ 𝑋))
29	5, 12, 15, 19, 28	syl13anc 1371	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → ((𝑃 ≤ 𝑋 ∧ 𝑅 ≤ 𝑋) ↔ (𝑃 ∨ 𝑅) ≤ 𝑋))
30	26, 27, 29	mpbi2and 709	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → (𝑃 ∨ 𝑅) ≤ 𝑋)
31	2, 3, 5, 9, 18, 19, 25, 30	lattrd 18409	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ 𝑋) → 𝑄 ≤ 𝑋)
32	31	3expia 1120	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃) → (𝑅 ≤ 𝑋 → 𝑄 ≤ 𝑋))
33	1, 32	mtod 197	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑅 ≠ 𝑃) → ¬ 𝑅 ≤ 𝑋)
34	33	ex 412	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑃 → ¬ 𝑅 ≤ 𝑋))
35	34	necon4ad 2958	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 → 𝑅 = 𝑃))
36		simp31 1208	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ 𝑋)
37		breq1 5151	. . 3 ⊢ (𝑅 = 𝑃 → (𝑅 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
38	36, 37	syl5ibrcom 246	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 = 𝑃 → 𝑅 ≤ 𝑋))
39	35, 38	impbid 211	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  Latclat 18394  Atomscatm 38600  HLchlt 38687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-covers 38603  df-ats 38604  df-atl 38635  df-cvlat 38659  df-hlat 38688

This theorem is referenced by:  cdlema2N  39130