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Metamath Proof Explorer

Theorem lplnllnneN

Description: Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012) (New usage is discouraged.)

Ref Expression
Hypotheses lplnri1.j 
|- .\/ = ( join ` K )
lplnri1.a 
|- A = ( Atoms ` K )
lplnri1.p 
|- P = ( LPlanes ` K )
lplnri1.y 
|- Y = ( ( Q .\/ R ) .\/ S )
Assertion lplnllnneN 
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( Q .\/ S ) =/= ( R .\/ S ) )

Proof

Step Hyp Ref Expression
1 lplnri1.j 
 |-  .\/ = ( join ` K )
2 lplnri1.a 
 |-  A = ( Atoms ` K )
3 lplnri1.p 
 |-  P = ( LPlanes ` K )
4 lplnri1.y 
 |-  Y = ( ( Q .\/ R ) .\/ S )
5 eqid 
 |-  ( le ` K ) = ( le ` K )
6 5 1 2 3 4 lplnriaN 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> -. Q ( le ` K ) ( R .\/ S ) )
7 simpl1 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> K e. HL )
8 simpl21 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> Q e. A )
9 simpl23 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> S e. A )
10 5 1 2 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ S e. A ) -> Q ( le ` K ) ( Q .\/ S ) )
11 7 8 9 10 syl3anc 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> Q ( le ` K ) ( Q .\/ S ) )
12 simpr 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> ( Q .\/ S ) = ( R .\/ S ) )
13 11 12 breqtrd 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> Q ( le ` K ) ( R .\/ S ) )
14 13 ex 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( ( Q .\/ S ) = ( R .\/ S ) -> Q ( le ` K ) ( R .\/ S ) ) )
15 14 necon3bd 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( -. Q ( le ` K ) ( R .\/ S ) -> ( Q .\/ S ) =/= ( R .\/ S ) ) )
16 6 15 mpd 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( Q .\/ S ) =/= ( R .\/ S ) )