Theorem cvlexchb1 36626

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 36622	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2	1	adantr 484	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ Lat)
3		simpr3 1193	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1192	. . . . . . . . 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 36585	. . . . . . . . 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 17662	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄))
12	2, 3, 8, 11	syl3anc 1368	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
13	12	3adant3 1129	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑄))
14	13	adantr 484	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
15		simpr 488	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
16		simpr1 1191	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴)
17	5, 6	atbase 36585	. . . . . . . . 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 17653	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
20	2, 3, 8, 19	syl3anc 1368	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
21	5, 9, 10	latjle12 17664	. . . . . . . 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 1369	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
23	22	3adant3 1129	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
24	23	adantr 484	. . . . 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 17662	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑃))
27	2, 3, 18, 26	syl3anc 1368	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
28	27	3adant3 1129	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑃))
29	28	adantr 484	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
30	5, 9, 10, 6	cvlexch1 36624	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
31	30	imp 410	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))
32	5, 10	latjcl 17653	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) ∈ 𝐵)
33	2, 3, 18, 32	syl3anc 1368	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑃) ∈ 𝐵)
34	5, 9, 10	latjle12 17664	. . . . . . . 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 1369	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
36	35	3adant3 1129	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
37	36	adantr 484	. . . . 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 17656	. . . . . . 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 1368	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
41	40	3adant3 1129	. . . . 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 484	. . . 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 416	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
45	5, 9, 10	latlej2 17663	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑃 ≤ (𝑋 ∨ 𝑃))
46	2, 3, 18, 45	syl3anc 1368	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ≤ (𝑋 ∨ 𝑃))
47		breq2 5034	. . . 4 ⊢ ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → (𝑃 ≤ (𝑋 ∨ 𝑃) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
48	46, 47	syl5ibcom 248	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
49	48	3adant3 1129	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
50	44, 49	impbid 215	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  Atomscatm 36559  CvLatclc 36561

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-lat 17648  df-ats 36563  df-atl 36594  df-cvlat 36618

This theorem is referenced by:  cvlexchb2  36627  cvlexch4N  36629  cvlatexchb1  36630  cvlcvr1  36635  hlexchb1  36680