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Metamath Proof Explorer

Theorem olm01

Description: Meet with lattice zero is zero. ( chm0 analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses olm0.b 
|- B = ( Base ` K )
olm0.m 
|- ./\ = ( meet ` K )
olm0.z 
|- .0. = ( 0. ` K )
Assertion olm01 
|- ( ( K e. OL /\ X e. B ) -> ( X ./\ .0. ) = .0. )

Proof

Step Hyp Ref Expression
1 olm0.b 
 |-  B = ( Base ` K )
2 olm0.m 
 |-  ./\ = ( meet ` K )
3 olm0.z 
 |-  .0. = ( 0. ` K )
4 eqid 
 |-  ( le ` K ) = ( le ` K )
5 ollat 
 |-  ( K e. OL -> K e. Lat )
6 5 adantr 
 |-  ( ( K e. OL /\ X e. B ) -> K e. Lat )
7 simpr 
 |-  ( ( K e. OL /\ X e. B ) -> X e. B )
8 olop 
 |-  ( K e. OL -> K e. OP )
9 8 adantr 
 |-  ( ( K e. OL /\ X e. B ) -> K e. OP )
10 1 3 op0cl 
 |-  ( K e. OP -> .0. e. B )
11 9 10 syl 
 |-  ( ( K e. OL /\ X e. B ) -> .0. e. B )
12 1 2 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ .0. e. B ) -> ( X ./\ .0. ) e. B )
13 6 7 11 12 syl3anc 
 |-  ( ( K e. OL /\ X e. B ) -> ( X ./\ .0. ) e. B )
14 1 4 2 latmle2 
 |-  ( ( K e. Lat /\ X e. B /\ .0. e. B ) -> ( X ./\ .0. ) ( le ` K ) .0. )
15 6 7 11 14 syl3anc 
 |-  ( ( K e. OL /\ X e. B ) -> ( X ./\ .0. ) ( le ` K ) .0. )
16 1 4 3 op0le 
 |-  ( ( K e. OP /\ X e. B ) -> .0. ( le ` K ) X )
17 8 16 sylan 
 |-  ( ( K e. OL /\ X e. B ) -> .0. ( le ` K ) X )
18 1 4 latref 
 |-  ( ( K e. Lat /\ .0. e. B ) -> .0. ( le ` K ) .0. )
19 6 11 18 syl2anc 
 |-  ( ( K e. OL /\ X e. B ) -> .0. ( le ` K ) .0. )
20 1 4 2 latlem12 
 |-  ( ( K e. Lat /\ ( .0. e. B /\ X e. B /\ .0. e. B ) ) -> ( ( .0. ( le ` K ) X /\ .0. ( le ` K ) .0. ) <-> .0. ( le ` K ) ( X ./\ .0. ) ) )
21 6 11 7 11 20 syl13anc 
 |-  ( ( K e. OL /\ X e. B ) -> ( ( .0. ( le ` K ) X /\ .0. ( le ` K ) .0. ) <-> .0. ( le ` K ) ( X ./\ .0. ) ) )
22 17 19 21 mpbi2and 
 |-  ( ( K e. OL /\ X e. B ) -> .0. ( le ` K ) ( X ./\ .0. ) )
23 1 4 6 13 11 15 22 latasymd 
 |-  ( ( K e. OL /\ X e. B ) -> ( X ./\ .0. ) = .0. )