Metamath Proof Explorer

Theorem 3dimlem2

Description: Lemma for 3dim1 . (Contributed by NM, 25-Jul-2012)

Ref Expression
Hypotheses 3dim0.j 
|- .\/ = ( join ` K )
3dim0.l 
|- .<_ = ( le ` K )
3dim0.a 
|- A = ( Atoms ` K )
Assertion 3dimlem2 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( ( P .\/ Q ) .\/ S ) ) )

Proof

Step Hyp Ref Expression
1 3dim0.j 
 |-  .\/ = ( join ` K )
2 3dim0.l 
 |-  .<_ = ( le ` K )
3 3dim0.a 
 |-  A = ( Atoms ` K )
4 simp3l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> P =/= Q )
5 simp22 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> -. S .<_ ( Q .\/ R ) )
6 1 3 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
7 6 3ad2ant1 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
8 simp3r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> P .<_ ( Q .\/ R ) )
9 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> K e. HL )
10 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> P e. A )
11 simp21 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> R e. A )
12 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> Q e. A )
13 2 1 3 hlatexchb1 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( Q .\/ R ) <-> ( Q .\/ P ) = ( Q .\/ R ) ) )
14 9 10 11 12 4 13 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( P .<_ ( Q .\/ R ) <-> ( Q .\/ P ) = ( Q .\/ R ) ) )
15 8 14 mpbid 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( Q .\/ P ) = ( Q .\/ R ) )
16 7 15 eqtrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( P .\/ Q ) = ( Q .\/ R ) )
17 16 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( S .<_ ( P .\/ Q ) <-> S .<_ ( Q .\/ R ) ) )
18 5 17 mtbird 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> -. S .<_ ( P .\/ Q ) )
19 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> -. T .<_ ( ( Q .\/ R ) .\/ S ) )
20 16 oveq1d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( ( P .\/ Q ) .\/ S ) = ( ( Q .\/ R ) .\/ S ) )
21 20 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( T .<_ ( ( P .\/ Q ) .\/ S ) <-> T .<_ ( ( Q .\/ R ) .\/ S ) ) )
22 19 21 mtbird 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> -. T .<_ ( ( P .\/ Q ) .\/ S ) )
23 4 18 22 3jca 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. S .<_ ( Q .\/ R ) /\ -. T .<_ ( ( Q .\/ R ) .\/ S ) ) /\ ( P =/= Q /\ P .<_ ( Q .\/ R ) ) ) -> ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( ( P .\/ Q ) .\/ S ) ) )