Metamath Proof Explorer


Theorem cvlexch4N

Description: An atomic covering lattice has the exchange property. Part of Definition 7.8 of MaedaMaeda p. 32. (Contributed by NM, 5-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cvlexch3.b B = Base K
cvlexch3.l ˙ = K
cvlexch3.j ˙ = join K
cvlexch3.m ˙ = meet K
cvlexch3.z 0 ˙ = 0. K
cvlexch3.a A = Atoms K
Assertion cvlexch4N K CvLat P A Q A X B P ˙ X = 0 ˙ P ˙ X ˙ Q X ˙ P = X ˙ Q

Proof

Step Hyp Ref Expression
1 cvlexch3.b B = Base K
2 cvlexch3.l ˙ = K
3 cvlexch3.j ˙ = join K
4 cvlexch3.m ˙ = meet K
5 cvlexch3.z 0 ˙ = 0. K
6 cvlexch3.a A = Atoms K
7 cvlatl K CvLat K AtLat
8 7 adantr K CvLat P A Q A X B K AtLat
9 simpr1 K CvLat P A Q A X B P A
10 simpr3 K CvLat P A Q A X B X B
11 1 2 4 5 6 atnle K AtLat P A X B ¬ P ˙ X P ˙ X = 0 ˙
12 8 9 10 11 syl3anc K CvLat P A Q A X B ¬ P ˙ X P ˙ X = 0 ˙
13 1 2 3 6 cvlexchb1 K CvLat P A Q A X B ¬ P ˙ X P ˙ X ˙ Q X ˙ P = X ˙ Q
14 13 3expia K CvLat P A Q A X B ¬ P ˙ X P ˙ X ˙ Q X ˙ P = X ˙ Q
15 12 14 sylbird K CvLat P A Q A X B P ˙ X = 0 ˙ P ˙ X ˙ Q X ˙ P = X ˙ Q
16 15 3impia K CvLat P A Q A X B P ˙ X = 0 ˙ P ˙ X ˙ Q X ˙ P = X ˙ Q