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. )