Theorem exatleN 39393

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 1256	. . . . 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 1285	. . . . . . . 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 39353	. . . . . . 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 1307	. . . . . . . 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 39278	. . . . . . . 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 1306	. . . . . . . . 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 39278	. . . . . . . . 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 1308	. . . . . . . . 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 39278	. . . . . . . . 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 18345	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑅 ∈ 𝐵) → (𝑃 ∨ 𝑅) ∈ 𝐵)
18	5, 12, 15, 17	syl3anc 1373	. . . . . . 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 1286	. . . . . . 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 1128	. . . . . . . . 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 1137	. . . . . . . . 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 1128	. . . . . . . 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 1311	. . . . . . . 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 39384	. . . . . . . 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 1309	. . . . . . . 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 1138	. . . . . . . 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 18356	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑅 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑅 ≤ 𝑋) ↔ (𝑃 ∨ 𝑅) ≤ 𝑋))
29	5, 12, 15, 19, 28	syl13anc 1374	. . . . . . . 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 712	. . . . . . 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 18352	. . . . . 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 1121	. . . . 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 198	. . . 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 2944	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 → 𝑅 = 𝑃))
36		simp31 1210	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ 𝑋)
37		breq1 5095	. . 3 ⊢ (𝑅 = 𝑃 → (𝑅 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
38	36, 37	syl5ibrcom 247	. 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 212	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  joincjn 18217  Latclat 18337  Atomscatm 39252  HLchlt 39339

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-lat 18338  df-covers 39255  df-ats 39256  df-atl 39287  df-cvlat 39311  df-hlat 39340

This theorem is referenced by:  cdlema2N  39781