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Metamath Proof Explorer

Theorem dalem38

Description: Lemma for dath . Plane Y belongs to the 3-dimensional volume G H I c . (Contributed by NM, 5-Aug-2012)

Ref Expression
Hypotheses dalem.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 ) ) ) ) )
dalem.l 
|- .<_ = ( le ` K )
dalem.j 
|- .\/ = ( join ` K )
dalem.a 
|- A = ( Atoms ` K )
dalem.ps 
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
dalem38.m 
|- ./\ = ( meet ` K )
dalem38.o 
|- O = ( LPlanes ` K )
dalem38.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem38.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem38.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem38.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem38.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
Assertion dalem38 
|- ( ( ph /\ Y = Z /\ ps ) -> Y .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )

Proof

Step Hyp Ref Expression
1 dalem.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 dalem.l 
 |-  .<_ = ( le ` K )
3 dalem.j 
 |-  .\/ = ( join ` K )
4 dalem.a 
 |-  A = ( Atoms ` K )
5 dalem.ps 
 |-  ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
6 dalem38.m 
 |-  ./\ = ( meet ` K )
7 dalem38.o 
 |-  O = ( LPlanes ` K )
8 dalem38.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem38.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem38.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 dalem38.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
12 dalem38.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
13 1 2 3 4 5 6 7 8 9 10 dalem28 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P .<_ ( G .\/ c ) )
14 1 2 3 4 5 6 7 8 9 11 dalem33 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Q .<_ ( H .\/ c ) )
15 1 dalemkelat 
 |-  ( ph -> K e. Lat )
16 15 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
17 1 4 dalempeb 
 |-  ( ph -> P e. ( Base ` K ) )
18 17 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P e. ( Base ` K ) )
19 1 dalemkehl 
 |-  ( ph -> K e. HL )
20 19 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
21 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
22 5 dalemccea 
 |-  ( ps -> c e. A )
23 22 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
24 eqid 
 |-  ( Base ` K ) = ( Base ` K )
25 24 3 4 hlatjcl 
 |-  ( ( K e. HL /\ G e. A /\ c e. A ) -> ( G .\/ c ) e. ( Base ` K ) )
26 20 21 23 25 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ c ) e. ( Base ` K ) )
27 1 4 dalemqeb 
 |-  ( ph -> Q e. ( Base ` K ) )
28 27 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Q e. ( Base ` K ) )
29 1 2 3 4 5 6 7 8 9 11 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
30 24 3 4 hlatjcl 
 |-  ( ( K e. HL /\ H e. A /\ c e. A ) -> ( H .\/ c ) e. ( Base ` K ) )
31 20 29 23 30 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( H .\/ c ) e. ( Base ` K ) )
32 24 2 3 latjlej12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ ( G .\/ c ) e. ( Base ` K ) ) /\ ( Q e. ( Base ` K ) /\ ( H .\/ c ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( G .\/ c ) /\ Q .<_ ( H .\/ c ) ) -> ( P .\/ Q ) .<_ ( ( G .\/ c ) .\/ ( H .\/ c ) ) ) )
33 16 18 26 28 31 32 syl122anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .<_ ( G .\/ c ) /\ Q .<_ ( H .\/ c ) ) -> ( P .\/ Q ) .<_ ( ( G .\/ c ) .\/ ( H .\/ c ) ) ) )
34 13 14 33 mp2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) .<_ ( ( G .\/ c ) .\/ ( H .\/ c ) ) )
35 24 4 atbase 
 |-  ( G e. A -> G e. ( Base ` K ) )
36 21 35 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. ( Base ` K ) )
37 24 4 atbase 
 |-  ( H e. A -> H e. ( Base ` K ) )
38 29 37 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. ( Base ` K ) )
39 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
40 39 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. ( Base ` K ) )
41 24 3 latjjdir 
 |-  ( ( K e. Lat /\ ( G e. ( Base ` K ) /\ H e. ( Base ` K ) /\ c e. ( Base ` K ) ) ) -> ( ( G .\/ H ) .\/ c ) = ( ( G .\/ c ) .\/ ( H .\/ c ) ) )
42 16 36 38 40 41 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ c ) = ( ( G .\/ c ) .\/ ( H .\/ c ) ) )
43 34 42 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) .<_ ( ( G .\/ H ) .\/ c ) )
44 1 2 3 4 5 6 7 8 9 12 dalem37 
 |-  ( ( ph /\ Y = Z /\ ps ) -> R .<_ ( I .\/ c ) )
45 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
46 45 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) e. ( Base ` K ) )
47 24 3 4 hlatjcl 
 |-  ( ( K e. HL /\ G e. A /\ H e. A ) -> ( G .\/ H ) e. ( Base ` K ) )
48 20 21 29 47 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( Base ` K ) )
49 24 3 latjcl 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ c e. ( Base ` K ) ) -> ( ( G .\/ H ) .\/ c ) e. ( Base ` K ) )
50 16 48 40 49 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ c ) e. ( Base ` K ) )
51 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
52 51 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> R e. ( Base ` K ) )
53 1 2 3 4 5 6 7 8 9 12 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
54 24 3 4 hlatjcl 
 |-  ( ( K e. HL /\ I e. A /\ c e. A ) -> ( I .\/ c ) e. ( Base ` K ) )
55 20 53 23 54 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( I .\/ c ) e. ( Base ` K ) )
56 24 2 3 latjlej12 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( ( G .\/ H ) .\/ c ) e. ( Base ` K ) ) /\ ( R e. ( Base ` K ) /\ ( I .\/ c ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .<_ ( ( G .\/ H ) .\/ c ) /\ R .<_ ( I .\/ c ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( ( G .\/ H ) .\/ c ) .\/ ( I .\/ c ) ) ) )
57 16 46 50 52 55 56 syl122anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( P .\/ Q ) .<_ ( ( G .\/ H ) .\/ c ) /\ R .<_ ( I .\/ c ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( ( G .\/ H ) .\/ c ) .\/ ( I .\/ c ) ) ) )
58 43 44 57 mp2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( ( G .\/ H ) .\/ c ) .\/ ( I .\/ c ) ) )
59 24 4 atbase 
 |-  ( I e. A -> I e. ( Base ` K ) )
60 53 59 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. ( Base ` K ) )
61 24 3 latjjdir 
 |-  ( ( K e. Lat /\ ( ( G .\/ H ) e. ( Base ` K ) /\ I e. ( Base ` K ) /\ c e. ( Base ` K ) ) ) -> ( ( ( G .\/ H ) .\/ I ) .\/ c ) = ( ( ( G .\/ H ) .\/ c ) .\/ ( I .\/ c ) ) )
62 16 48 60 40 61 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) .\/ I ) .\/ c ) = ( ( ( G .\/ H ) .\/ c ) .\/ ( I .\/ c ) ) )
63 58 62 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )
64 8 63 eqbrtrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )