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Metamath Proof Explorer

Theorem dalem24

Description: Lemma for dath . Show that auxiliary atom G is outside of plane Y . (Contributed by NM, 2-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 ) ) ) )
dalem23.m 
|- ./\ = ( meet ` K )
dalem23.o 
|- O = ( LPlanes ` K )
dalem23.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem23.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem23.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
Assertion dalem24 
|- ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )

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 dalem23.m 
 |-  ./\ = ( meet ` K )
7 dalem23.o 
 |-  O = ( LPlanes ` K )
8 dalem23.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem23.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem23.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 10 oveq1i 
 |-  ( G ./\ Y ) = ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y )
12 1 dalemkehl 
 |-  ( ph -> K e. HL )
13 hlol 
 |-  ( K e. HL -> K e. OL )
14 12 13 syl 
 |-  ( ph -> K e. OL )
15 14 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. OL )
16 12 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
17 5 dalemccea 
 |-  ( ps -> c e. A )
18 17 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
19 1 dalempea 
 |-  ( ph -> P e. A )
20 19 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P e. A )
21 eqid 
 |-  ( Base ` K ) = ( Base ` K )
22 21 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
23 16 18 20 22 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
24 5 dalemddea 
 |-  ( ps -> d e. A )
25 24 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d e. A )
26 1 dalemsea 
 |-  ( ph -> S e. A )
27 26 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S e. A )
28 21 3 4 hlatjcl 
 |-  ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
29 16 25 27 28 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
30 1 7 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
31 30 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
32 21 6 latmmdir 
 |-  ( ( K e. OL /\ ( ( c .\/ P ) e. ( Base ` K ) /\ ( d .\/ S ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
33 15 23 29 31 32 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
34 11 33 eqtrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) )
35 3 4 hlatjcom 
 |-  ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) = ( P .\/ c ) )
36 16 18 20 35 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) = ( P .\/ c ) )
37 36 oveq1d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ Y ) = ( ( P .\/ c ) ./\ Y ) )
38 1 2 3 4 7 8 dalemply 
 |-  ( ph -> P .<_ Y )
39 38 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P .<_ Y )
40 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
41 40 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
42 21 2 3 6 4 2atjm 
 |-  ( ( K e. HL /\ ( P e. A /\ c e. A /\ Y e. ( Base ` K ) ) /\ ( P .<_ Y /\ -. c .<_ Y ) ) -> ( ( P .\/ c ) ./\ Y ) = P )
43 16 20 18 31 39 41 42 syl132anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ c ) ./\ Y ) = P )
44 37 43 eqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ Y ) = P )
45 3 4 hlatjcom 
 |-  ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) = ( S .\/ d ) )
46 16 25 27 45 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) = ( S .\/ d ) )
47 46 oveq1d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( d .\/ S ) ./\ Y ) = ( ( S .\/ d ) ./\ Y ) )
48 1 2 3 4 9 dalemsly 
 |-  ( ( ph /\ Y = Z ) -> S .<_ Y )
49 48 3adant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
50 5 dalem-ddly 
 |-  ( ps -> -. d .<_ Y )
51 50 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
52 21 2 3 6 4 2atjm 
 |-  ( ( K e. HL /\ ( S e. A /\ d e. A /\ Y e. ( Base ` K ) ) /\ ( S .<_ Y /\ -. d .<_ Y ) ) -> ( ( S .\/ d ) ./\ Y ) = S )
53 16 27 25 31 49 51 52 syl132anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( S .\/ d ) ./\ Y ) = S )
54 47 53 eqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( d .\/ S ) ./\ Y ) = S )
55 44 54 oveq12d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( c .\/ P ) ./\ Y ) ./\ ( ( d .\/ S ) ./\ Y ) ) = ( P ./\ S ) )
56 1 2 3 4 7 8 dalempnes 
 |-  ( ph -> P =/= S )
57 hlatl 
 |-  ( K e. HL -> K e. AtLat )
58 12 57 syl 
 |-  ( ph -> K e. AtLat )
59 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
60 6 59 4 atnem0 
 |-  ( ( K e. AtLat /\ P e. A /\ S e. A ) -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
61 58 19 26 60 syl3anc 
 |-  ( ph -> ( P =/= S <-> ( P ./\ S ) = ( 0. ` K ) ) )
62 56 61 mpbid 
 |-  ( ph -> ( P ./\ S ) = ( 0. ` K ) )
63 62 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P ./\ S ) = ( 0. ` K ) )
64 34 55 63 3eqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G ./\ Y ) = ( 0. ` K ) )
65 58 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
66 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
67 21 2 6 59 4 atnle 
 |-  ( ( K e. AtLat /\ G e. A /\ Y e. ( Base ` K ) ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
68 65 66 31 67 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( -. G .<_ Y <-> ( G ./\ Y ) = ( 0. ` K ) ) )
69 64 68 mpbird 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )