This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem iscvlat2N

Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses iscvlat2.b 
|- B = ( Base ` K )
iscvlat2.l 
|- .<_ = ( le ` K )
iscvlat2.j 
|- .\/ = ( join ` K )
iscvlat2.m 
|- ./\ = ( meet ` K )
iscvlat2.z 
|- .0. = ( 0. ` K )
iscvlat2.a 
|- A = ( Atoms ` K )
Assertion iscvlat2N 
|- ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )

Proof

Step Hyp Ref Expression
1 iscvlat2.b 
 |-  B = ( Base ` K )
2 iscvlat2.l 
 |-  .<_ = ( le ` K )
3 iscvlat2.j 
 |-  .\/ = ( join ` K )
4 iscvlat2.m 
 |-  ./\ = ( meet ` K )
5 iscvlat2.z 
 |-  .0. = ( 0. ` K )
6 iscvlat2.a 
 |-  A = ( Atoms ` K )
7 1 2 3 6 iscvlat 
 |-  ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
8 simpll 
 |-  ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> K e. AtLat )
9 simplrl 
 |-  ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> p e. A )
10 simpr 
 |-  ( ( ( K e. AtLat /\ ( 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. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> ( -. p .<_ x <-> ( p ./\ x ) = .0. ) )
13 12 anbi1d 
 |-  ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) <-> ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) ) )
14 13 imbi1d 
 |-  ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> ( ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) <-> ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
15 14 ralbidva 
 |-  ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) -> ( A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) <-> A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
16 15 2ralbidva 
 |-  ( K e. AtLat -> ( A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) <-> A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
17 16 pm5.32i 
 |-  ( ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
18 7 17 bitri 
 |-  ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )