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Metamath Proof Explorer

Theorem 2llnm4

Description: Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012)

Ref Expression
Hypotheses 2llnm4.l 
|- .<_ = ( le ` K )
2llnm4.m 
|- ./\ = ( meet ` K )
2llnm4.z 
|- .0. = ( 0. ` K )
2llnm4.a 
|- A = ( Atoms ` K )
2llnm4.n 
|- N = ( LLines ` K )
Assertion 2llnm4 
|- ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> ( X ./\ Y ) =/= .0. )

Proof

Step Hyp Ref Expression
1 2llnm4.l 
 |-  .<_ = ( le ` K )
2 2llnm4.m 
 |-  ./\ = ( meet ` K )
3 2llnm4.z 
 |-  .0. = ( 0. ` K )
4 2llnm4.a 
 |-  A = ( Atoms ` K )
5 2llnm4.n 
 |-  N = ( LLines ` K )
6 hlatl 
 |-  ( K e. HL -> K e. AtLat )
7 6 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> K e. AtLat )
8 hllat 
 |-  ( K e. HL -> K e. Lat )
9 8 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> K e. Lat )
10 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> X e. N )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 5 llnbase 
 |-  ( X e. N -> X e. ( Base ` K ) )
13 10 12 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> X e. ( Base ` K ) )
14 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> Y e. N )
15 11 5 llnbase 
 |-  ( Y e. N -> Y e. ( Base ` K ) )
16 14 15 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> Y e. ( Base ` K ) )
17 11 2 latmcl 
 |-  ( ( K e. Lat /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( X ./\ Y ) e. ( Base ` K ) )
18 9 13 16 17 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> ( X ./\ Y ) e. ( Base ` K ) )
19 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> P e. A )
20 simp3 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> ( P .<_ X /\ P .<_ Y ) )
21 11 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
22 19 21 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> P e. ( Base ` K ) )
23 11 1 2 latlem12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( P .<_ X /\ P .<_ Y ) <-> P .<_ ( X ./\ Y ) ) )
24 9 22 13 16 23 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> ( ( P .<_ X /\ P .<_ Y ) <-> P .<_ ( X ./\ Y ) ) )
25 20 24 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> P .<_ ( X ./\ Y ) )
26 11 1 3 4 atlen0 
 |-  ( ( ( K e. AtLat /\ ( X ./\ Y ) e. ( Base ` K ) /\ P e. A ) /\ P .<_ ( X ./\ Y ) ) -> ( X ./\ Y ) =/= .0. )
27 7 18 19 25 26 syl31anc 
 |-  ( ( K e. HL /\ ( P e. A /\ X e. N /\ Y e. N ) /\ ( P .<_ X /\ P .<_ Y ) ) -> ( X ./\ Y ) =/= .0. )