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Metamath Proof Explorer

Theorem dalem5

Description: Lemma for dath . Atom U (in plane Z = S T U ) belongs to the 3-dimensional volume formed by Y and C . (Contributed by NM, 21-Jul-2012)

Ref Expression
Hypotheses dalema.ph 
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalemc.l 
|- .<_ = ( le ` K )
dalemc.j 
|- .\/ = ( join ` K )
dalemc.a 
|- A = ( Atoms ` K )
dalem5.o 
|- O = ( LPlanes ` K )
dalem5.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem5.w 
|- W = ( Y .\/ C )
Assertion dalem5 
|- ( ph -> U .<_ W )

Proof

Step Hyp Ref Expression
1 dalema.ph 
 |-  ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
2 dalemc.l 
 |-  .<_ = ( le ` K )
3 dalemc.j 
 |-  .\/ = ( join ` K )
4 dalemc.a 
 |-  A = ( Atoms ` K )
5 dalem5.o 
 |-  O = ( LPlanes ` K )
6 dalem5.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 dalem5.w 
 |-  W = ( Y .\/ C )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 1 dalemkelat 
 |-  ( ph -> K e. Lat )
10 1 4 dalemueb 
 |-  ( ph -> U e. ( Base ` K ) )
11 1 dalemkehl 
 |-  ( ph -> K e. HL )
12 1 dalemrea 
 |-  ( ph -> R e. A )
13 1 2 3 4 5 6 dalemcea 
 |-  ( ph -> C e. A )
14 8 3 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ C e. A ) -> ( R .\/ C ) e. ( Base ` K ) )
15 11 12 13 14 syl3anc 
 |-  ( ph -> ( R .\/ C ) e. ( Base ` K ) )
16 1 5 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
17 1 4 dalemceb 
 |-  ( ph -> C e. ( Base ` K ) )
18 8 3 latjcl 
 |-  ( ( K e. Lat /\ Y e. ( Base ` K ) /\ C e. ( Base ` K ) ) -> ( Y .\/ C ) e. ( Base ` K ) )
19 9 16 17 18 syl3anc 
 |-  ( ph -> ( Y .\/ C ) e. ( Base ` K ) )
20 7 19 eqeltrid 
 |-  ( ph -> W e. ( Base ` K ) )
21 1 dalemclrju 
 |-  ( ph -> C .<_ ( R .\/ U ) )
22 1 dalemuea 
 |-  ( ph -> U e. A )
23 1 dalempea 
 |-  ( ph -> P e. A )
24 simp313 
 |-  ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) -> -. C .<_ ( R .\/ P ) )
25 1 24 sylbi 
 |-  ( ph -> -. C .<_ ( R .\/ P ) )
26 2 3 4 atnlej1 
 |-  ( ( K e. HL /\ ( C e. A /\ R e. A /\ P e. A ) /\ -. C .<_ ( R .\/ P ) ) -> C =/= R )
27 11 13 12 23 25 26 syl131anc 
 |-  ( ph -> C =/= R )
28 2 3 4 hlatexch1 
 |-  ( ( K e. HL /\ ( C e. A /\ U e. A /\ R e. A ) /\ C =/= R ) -> ( C .<_ ( R .\/ U ) -> U .<_ ( R .\/ C ) ) )
29 11 13 22 12 27 28 syl131anc 
 |-  ( ph -> ( C .<_ ( R .\/ U ) -> U .<_ ( R .\/ C ) ) )
30 21 29 mpd 
 |-  ( ph -> U .<_ ( R .\/ C ) )
31 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
32 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
33 8 2 3 latlej2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> R .<_ ( ( P .\/ Q ) .\/ R ) )
34 9 31 32 33 syl3anc 
 |-  ( ph -> R .<_ ( ( P .\/ Q ) .\/ R ) )
35 34 6 breqtrrdi 
 |-  ( ph -> R .<_ Y )
36 8 2 3 latjlej1 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ Y e. ( Base ` K ) /\ C e. ( Base ` K ) ) ) -> ( R .<_ Y -> ( R .\/ C ) .<_ ( Y .\/ C ) ) )
37 9 32 16 17 36 syl13anc 
 |-  ( ph -> ( R .<_ Y -> ( R .\/ C ) .<_ ( Y .\/ C ) ) )
38 35 37 mpd 
 |-  ( ph -> ( R .\/ C ) .<_ ( Y .\/ C ) )
39 38 7 breqtrrdi 
 |-  ( ph -> ( R .\/ C ) .<_ W )
40 8 2 9 10 15 20 30 39 lattrd 
 |-  ( ph -> U .<_ W )