Metamath Proof Explorer

Theorem dalem39

Description: Lemma for dath . Auxiliary atoms G , H , and I are not colinear. (Contributed by NM, 4-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 dalem39 
|- ( ( ph /\ Y = Z /\ ps ) -> -. H .<_ ( I .\/ G ) )

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 dalemkehl 
 |-  ( ph -> K e. HL )
14 13 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
15 1 dalemyeo 
 |-  ( ph -> Y e. O )
16 15 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. O )
17 5 dalemccea 
 |-  ( ps -> c e. A )
18 17 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
19 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
20 19 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
21 eqid 
 |-  ( LVols ` K ) = ( LVols ` K )
22 2 3 4 7 21 lvoli3 
 |-  ( ( ( K e. HL /\ Y e. O /\ c e. A ) /\ -. c .<_ Y ) -> ( Y .\/ c ) e. ( LVols ` K ) )
23 14 16 18 20 22 syl31anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( Y .\/ c ) e. ( LVols ` K ) )
24 1 2 3 4 5 6 7 8 9 12 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
25 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
26 2 3 4 21 lvolnle3at 
 |-  ( ( ( K e. HL /\ ( Y .\/ c ) e. ( LVols ` K ) ) /\ ( I e. A /\ G e. A /\ c e. A ) ) -> -. ( Y .\/ c ) .<_ ( ( I .\/ G ) .\/ c ) )
27 14 23 24 25 18 26 syl23anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. ( Y .\/ c ) .<_ ( ( I .\/ G ) .\/ c ) )
28 1 2 3 4 5 6 7 8 9 10 11 12 dalem38 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )
29 1 dalemkelat 
 |-  ( ph -> K e. Lat )
30 29 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
31 1 2 3 4 5 6 7 8 9 11 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
32 eqid 
 |-  ( Base ` K ) = ( Base ` K )
33 32 3 4 hlatjcl 
 |-  ( ( K e. HL /\ G e. A /\ H e. A ) -> ( G .\/ H ) e. ( Base ` K ) )
34 14 25 31 33 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( Base ` K ) )
35 32 4 atbase 
 |-  ( I e. A -> I e. ( Base ` K ) )
36 24 35 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. ( Base ` K ) )
37 32 3 latjcl 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ I e. ( Base ` K ) ) -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
38 30 34 36 37 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
39 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
40 39 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. ( Base ` K ) )
41 32 2 3 latlej2 
 |-  ( ( K e. Lat /\ ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) /\ c e. ( Base ` K ) ) -> c .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )
42 30 38 40 41 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )
43 1 7 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
44 43 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
45 32 3 latjcl 
 |-  ( ( K e. Lat /\ ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) /\ c e. ( Base ` K ) ) -> ( ( ( G .\/ H ) .\/ I ) .\/ c ) e. ( Base ` K ) )
46 30 38 40 45 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) .\/ I ) .\/ c ) e. ( Base ` K ) )
47 32 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( Y e. ( Base ` K ) /\ c e. ( Base ` K ) /\ ( ( ( G .\/ H ) .\/ I ) .\/ c ) e. ( Base ` K ) ) ) -> ( ( Y .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) /\ c .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) ) <-> ( Y .\/ c ) .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) ) )
48 30 44 40 46 47 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( Y .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) /\ c .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) ) <-> ( Y .\/ c ) .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) ) )
49 28 42 48 mpbi2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( Y .\/ c ) .<_ ( ( ( G .\/ H ) .\/ I ) .\/ c ) )
50 3 4 hlatjrot 
 |-  ( ( K e. HL /\ ( G e. A /\ H e. A /\ I e. A ) ) -> ( ( G .\/ H ) .\/ I ) = ( ( I .\/ G ) .\/ H ) )
51 14 25 31 24 50 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) = ( ( I .\/ G ) .\/ H ) )
52 51 oveq1d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) .\/ I ) .\/ c ) = ( ( ( I .\/ G ) .\/ H ) .\/ c ) )
53 49 52 breqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( Y .\/ c ) .<_ ( ( ( I .\/ G ) .\/ H ) .\/ c ) )
54 53 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ H .<_ ( I .\/ G ) ) -> ( Y .\/ c ) .<_ ( ( ( I .\/ G ) .\/ H ) .\/ c ) )
55 32 4 atbase 
 |-  ( H e. A -> H e. ( Base ` K ) )
56 31 55 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. ( Base ` K ) )
57 32 3 4 hlatjcl 
 |-  ( ( K e. HL /\ I e. A /\ G e. A ) -> ( I .\/ G ) e. ( Base ` K ) )
58 14 24 25 57 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( I .\/ G ) e. ( Base ` K ) )
59 32 2 3 latleeqj2 
 |-  ( ( K e. Lat /\ H e. ( Base ` K ) /\ ( I .\/ G ) e. ( Base ` K ) ) -> ( H .<_ ( I .\/ G ) <-> ( ( I .\/ G ) .\/ H ) = ( I .\/ G ) ) )
60 30 56 58 59 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( H .<_ ( I .\/ G ) <-> ( ( I .\/ G ) .\/ H ) = ( I .\/ G ) ) )
61 60 biimpa 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ H .<_ ( I .\/ G ) ) -> ( ( I .\/ G ) .\/ H ) = ( I .\/ G ) )
62 61 oveq1d 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ H .<_ ( I .\/ G ) ) -> ( ( ( I .\/ G ) .\/ H ) .\/ c ) = ( ( I .\/ G ) .\/ c ) )
63 54 62 breqtrd 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ H .<_ ( I .\/ G ) ) -> ( Y .\/ c ) .<_ ( ( I .\/ G ) .\/ c ) )
64 27 63 mtand 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. H .<_ ( I .\/ G ) )