Theorem cvlexchb1 39353

Description: An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
cvlexchb1	⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))

Proof of Theorem cvlexchb1

Step	Hyp	Ref	Expression
1		cvllat 39349	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ Lat)
3		simpr3 1197	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1196	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑄 ∈ 𝐴)
5		cvlexch.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
6		cvlexch.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	atbase 39312	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
8	4, 7	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑄 ∈ 𝐵)
9		cvlexch.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
10		cvlexch.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
11	5, 9, 10	latlej1 18463	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄))
12	2, 3, 8, 11	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
13	12	3adant3 1132	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑄))
14	13	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
15		simpr 484	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
16		simpr1 1195	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴)
17	5, 6	atbase 39312	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐵)
19	5, 10	latjcl 18454	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
20	2, 3, 8, 19	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
21	5, 9, 10	latjle12 18465	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
22	2, 3, 18, 20, 21	syl13anc 1374	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
23	22	3adant3 1132	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
24	23	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
25	14, 15, 24	mpbi2and 712	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄))
26	5, 9, 10	latlej1 18463	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑃))
27	2, 3, 18, 26	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
28	27	3adant3 1132	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑃))
29	28	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
30	5, 9, 10, 6	cvlexch1 39351	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
31	30	imp 406	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))
32	5, 10	latjcl 18454	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) ∈ 𝐵)
33	2, 3, 18, 32	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑃) ∈ 𝐵)
34	5, 9, 10	latjle12 18465	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ (𝑋 ∨ 𝑃) ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
35	2, 3, 8, 33, 34	syl13anc 1374	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
36	35	3adant3 1132	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
37	36	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
38	29, 31, 37	mpbi2and 712	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃))
39	5, 9	latasymb 18457	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑃) ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
40	2, 33, 20, 39	syl3anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
41	40	3adant3 1132	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
42	41	adantr 480	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
43	25, 38, 42	mpbi2and 712	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))
44	43	ex 412	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
45	5, 9, 10	latlej2 18464	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑃 ≤ (𝑋 ∨ 𝑃))
46	2, 3, 18, 45	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ≤ (𝑋 ∨ 𝑃))
47		breq2 5128	. . . 4 ⊢ ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → (𝑃 ≤ (𝑋 ∨ 𝑃) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
48	46, 47	syl5ibcom 245	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
49	48	3adant3 1132	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
50	44, 49	impbid 212	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  joincjn 18328  Latclat 18446  Atomscatm 39286  CvLatclc 39288

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18311  df-poset 18330  df-lub 18361  df-glb 18362  df-join 18363  df-meet 18364  df-lat 18447  df-ats 39290  df-atl 39321  df-cvlat 39345

This theorem is referenced by:  cvlexchb2  39354  cvlexch4N  39356  cvlatexchb1  39357  cvlcvr1  39362  hlexchb1  39408