Metamath Proof Explorer

Theorem cvlexch3

Description: An atomic covering lattice has the exchange property. ( atexch analog.) (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlexch3.b 
|- B = ( Base ` K )
cvlexch3.l 
|- .<_ = ( le ` K )
cvlexch3.j 
|- .\/ = ( join ` K )
cvlexch3.m 
|- ./\ = ( meet ` K )
cvlexch3.z 
|- .0. = ( 0. ` K )
cvlexch3.a 
|- A = ( Atoms ` K )
Assertion cvlexch3 
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) )

Proof

Step Hyp Ref Expression
1 cvlexch3.b 
 |-  B = ( Base ` K )
2 cvlexch3.l 
 |-  .<_ = ( le ` 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 e. CvLat -> K e. AtLat )
8 7 adantr 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> K e. AtLat )
9 simpr1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> P e. A )
10 simpr3 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> X e. B )
11 1 2 4 5 6 atnle 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( -. P .<_ X <-> ( P ./\ X ) = .0. ) )
12 8 9 10 11 syl3anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( -. P .<_ X <-> ( P ./\ X ) = .0. ) )
13 1 2 3 6 cvlexch1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) )
14 13 3expia 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( -. P .<_ X -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) ) )
15 12 14 sylbird 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( ( P ./\ X ) = .0. -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) ) )
16 15 3impia 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) )