Metamath Proof Explorer


Theorem atlen0

Description: A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses atlen0.b
|- B = ( Base ` K )
atlen0.l
|- .<_ = ( le ` K )
atlen0.z
|- .0. = ( 0. ` K )
atlen0.a
|- A = ( Atoms ` K )
Assertion atlen0
|- ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> X =/= .0. )

Proof

Step Hyp Ref Expression
1 atlen0.b
 |-  B = ( Base ` K )
2 atlen0.l
 |-  .<_ = ( le ` K )
3 atlen0.z
 |-  .0. = ( 0. ` K )
4 atlen0.a
 |-  A = ( Atoms ` K )
5 simpl1
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> K e. AtLat )
6 1 3 atl0cl
 |-  ( K e. AtLat -> .0. e. B )
7 5 6 syl
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> .0. e. B )
8 simpl2
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> X e. B )
9 5 7 8 3jca
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> ( K e. AtLat /\ .0. e. B /\ X e. B ) )
10 simpl3
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> P e. A )
11 1 4 atbase
 |-  ( P e. A -> P e. B )
12 10 11 syl
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> P e. B )
13 eqid
 |-  ( 
14 3 13 4 atcvr0
 |-  ( ( K e. AtLat /\ P e. A ) -> .0. ( 
15 5 10 14 syl2anc
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> .0. ( 
16 eqid
 |-  ( lt ` K ) = ( lt ` K )
17 1 16 13 cvrlt
 |-  ( ( ( K e. AtLat /\ .0. e. B /\ P e. B ) /\ .0. (  .0. ( lt ` K ) P )
18 5 7 12 15 17 syl31anc
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> .0. ( lt ` K ) P )
19 simpr
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> P .<_ X )
20 atlpos
 |-  ( K e. AtLat -> K e. Poset )
21 5 20 syl
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> K e. Poset )
22 1 2 16 pltletr
 |-  ( ( K e. Poset /\ ( .0. e. B /\ P e. B /\ X e. B ) ) -> ( ( .0. ( lt ` K ) P /\ P .<_ X ) -> .0. ( lt ` K ) X ) )
23 21 7 12 8 22 syl13anc
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> ( ( .0. ( lt ` K ) P /\ P .<_ X ) -> .0. ( lt ` K ) X ) )
24 18 19 23 mp2and
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> .0. ( lt ` K ) X )
25 16 pltne
 |-  ( ( K e. AtLat /\ .0. e. B /\ X e. B ) -> ( .0. ( lt ` K ) X -> .0. =/= X ) )
26 9 24 25 sylc
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> .0. =/= X )
27 26 necomd
 |-  ( ( ( K e. AtLat /\ X e. B /\ P e. A ) /\ P .<_ X ) -> X =/= .0. )