This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 4at2

Description: Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4at2 
|- ( ( ( ( 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 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 ) ) ) )
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 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
9 8 3ad2ant1 
 |-  ( ( ( 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 ) e. ( Base ` K ) )
10 simp21 
 |-  ( ( ( 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 ) ) -> R e. A )
11 7 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
12 10 11 syl 
 |-  ( ( ( 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 ) ) -> R e. ( Base ` K ) )
13 simp22 
 |-  ( ( ( 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 ) ) -> S e. A )
14 7 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( ( 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 ) ) -> S e. ( Base ` K ) )
16 7 2 latjass 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
17 6 9 12 15 16 syl13anc 
 |-  ( ( ( 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 ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
18 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 )
19 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 )
20 7 2 3 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
21 5 18 19 20 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 ) )
22 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 )
23 7 3 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
24 22 23 syl 
 |-  ( ( ( 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. ( Base ` K ) )
25 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 )
26 7 3 atbase 
 |-  ( W e. A -> W e. ( Base ` K ) )
27 25 26 syl 
 |-  ( ( ( 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. ( Base ` K ) )
28 7 2 latjass 
 |-  ( ( K e. Lat /\ ( ( T .\/ U ) e. ( Base ` K ) /\ V e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( T .\/ U ) .\/ V ) .\/ W ) = ( ( T .\/ U ) .\/ ( V .\/ W ) ) )
29 6 21 24 27 28 syl13anc 
 |-  ( ( ( 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 ) ) )
30 17 29 breq12d 
 |-  ( ( ( 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 ) ) ) )
31 30 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 ) ) ) )
32 17 29 eqeq12d 
 |-  ( ( ( 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 ) ) ) )
33 32 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 ) ) ) )
34 4 31 33 3bitr4d 
 |-  ( ( ( ( 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 ) ) )