Metamath Proof Explorer

Theorem llnexch2N

Description: Line exchange property (compare cvlatexch2 for atoms). (Contributed by NM, 18-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses llnexch.l 
|- .<_ = ( le ` K )
llnexch.j 
|- .\/ = ( join ` K )
llnexch.m 
|- ./\ = ( meet ` K )
llnexch.a 
|- A = ( Atoms ` K )
llnexch.n 
|- N = ( LLines ` K )
Assertion llnexch2N 
|- ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> ( ( X ./\ Y ) .<_ Z -> ( X ./\ Z ) .<_ Y ) )

Proof

Step Hyp Ref Expression
1 llnexch.l 
 |-  .<_ = ( le ` K )
2 llnexch.j 
 |-  .\/ = ( join ` K )
3 llnexch.m 
 |-  ./\ = ( meet ` K )
4 llnexch.a 
 |-  A = ( Atoms ` K )
5 llnexch.n 
 |-  N = ( LLines ` K )
6 1 2 3 4 5 llnexchb2 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> ( ( X ./\ Y ) .<_ Z <-> ( X ./\ Y ) = ( X ./\ Z ) ) )
7 hllat 
 |-  ( K e. HL -> K e. Lat )
8 7 3ad2ant1 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> K e. Lat )
9 simp21 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> X e. N )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 5 llnbase 
 |-  ( X e. N -> X e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> X e. ( Base ` K ) )
13 simp22 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> Y e. N )
14 10 5 llnbase 
 |-  ( Y e. N -> Y e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> Y e. ( Base ` K ) )
16 10 1 3 latmle2 
 |-  ( ( K e. Lat /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( X ./\ Y ) .<_ Y )
17 8 12 15 16 syl3anc 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> ( X ./\ Y ) .<_ Y )
18 breq1 
 |-  ( ( X ./\ Y ) = ( X ./\ Z ) -> ( ( X ./\ Y ) .<_ Y <-> ( X ./\ Z ) .<_ Y ) )
19 17 18 syl5ibcom 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> ( ( X ./\ Y ) = ( X ./\ Z ) -> ( X ./\ Z ) .<_ Y ) )
20 6 19 sylbid 
 |-  ( ( K e. HL /\ ( X e. N /\ Y e. N /\ Z e. N ) /\ ( ( X ./\ Y ) e. A /\ X =/= Z ) ) -> ( ( X ./\ Y ) .<_ Z -> ( X ./\ Z ) .<_ Y ) )