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
|- .<_ = ( le ` K )
cvlexch3.j
|- .\/ = ( join ` K )
cvlexch3.m
|- ./\ = ( meet ` K )
cvlexch3.z
|- .0. = ( 0. ` K )
cvlexch3.a
|- A = ( Atoms ` K )
Assertion cvlexch4N
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. 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
 |-  .<_ = ( 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 cvlexchb1
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) )
14 13 3expia
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( -. P .<_ X -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) )
15 12 14 sylbird
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) ) -> ( ( P ./\ X ) = .0. -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) )
16 15 3impia
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) )