Metamath Proof Explorer

Theorem 4at

Description: Four atoms determine a lattice volume uniquely. Three-dimensional analogue of ps-1 and 3at . (Contributed by NM, 11-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4at 
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 1 2 3 4atlem12 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) )
5 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> K e. HL )
6 5 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> K e. Lat )
7 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> T e. A )
8 simp31 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> U e. A )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
11 5 7 8 10 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( T .\/ U ) e. ( Base ` K ) )
12 simp32 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> V e. A )
13 simp33 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> W e. A )
14 9 2 3 hlatjcl 
 |-  ( ( K e. HL /\ V e. A /\ W e. A ) -> ( V .\/ W ) e. ( Base ` K ) )
15 5 12 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( V .\/ W ) e. ( Base ` K ) )
16 9 2 latjcl 
 |-  ( ( K e. Lat /\ ( T .\/ U ) e. ( Base ` K ) /\ ( V .\/ W ) e. ( Base ` K ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) )
17 6 11 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) )
18 9 1 latref 
 |-  ( ( K e. Lat /\ ( ( T .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) )
19 6 17 18 syl2anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( ( T .\/ U ) .\/ ( V .\/ W ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) )
20 breq1 
 |-  ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( T .\/ U ) .\/ ( V .\/ W ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) )
21 19 20 syl5ibrcom 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) )
22 21 adantr 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) )
23 4 22 impbid 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) ) )