Theorem cvlexchb1 39286

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 39282	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ Lat)
3		simpr3 1196	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1195	. . . . . . . . 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 39245	. . . . . . . . 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 18518	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄))
12	2, 3, 8, 11	syl3anc 1371	. . . . . . 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 1194	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴)
17	5, 6	atbase 39245	. . . . . . . . 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 18509	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
20	2, 3, 8, 19	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
21	5, 9, 10	latjle12 18520	. . . . . . . 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 1372	. . . . . . 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 711	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄))
26	5, 9, 10	latlej1 18518	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑃))
27	2, 3, 18, 26	syl3anc 1371	. . . . . . 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 39284	. . . . . 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 18509	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) ∈ 𝐵)
33	2, 3, 18, 32	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑃) ∈ 𝐵)
34	5, 9, 10	latjle12 18520	. . . . . . . 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 1372	. . . . . . 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 711	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃))
39	5, 9	latasymb 18512	. . . . . . 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 1371	. . . . . 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 711	. . 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 18519	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑃 ≤ (𝑋 ∨ 𝑃))
46	2, 3, 18, 45	syl3anc 1371	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ≤ (𝑋 ∨ 𝑃))
47		breq2 5170	. . . 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 1087   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  Latclat 18501  Atomscatm 39219  CvLatclc 39221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-lat 18502  df-ats 39223  df-atl 39254  df-cvlat 39278

This theorem is referenced by:  cvlexchb2  39287  cvlexch4N  39289  cvlatexchb1  39290  cvlcvr1  39295  hlexchb1  39341