Metamath Proof Explorer

Theorem dalem44

Description: Lemma for dath . Dummy center of perspectivity c lies outside of plane G H I . (Contributed by NM, 16-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 ) ) ) )
dalem44.m 
|- ./\ = ( meet ` K )
dalem44.o 
|- O = ( LPlanes ` K )
dalem44.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem44.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem44.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem44.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem44.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
Assertion dalem44 
|- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ ( ( G .\/ H ) .\/ I ) )

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 dalem44.m 
 |-  ./\ = ( meet ` K )
7 dalem44.o 
 |-  O = ( LPlanes ` K )
8 dalem44.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem44.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem44.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 dalem44.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
12 dalem44.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
13 1 2 3 4 5 6 7 8 9 10 11 12 dalem43 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) =/= Y )
14 13 necomd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y =/= ( ( G .\/ H ) .\/ I ) )
15 1 dalemkelat 
 |-  ( ph -> K e. Lat )
16 15 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
17 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
18 17 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. ( Base ` K ) )
19 1 2 3 4 5 6 7 8 9 10 11 12 dalem42 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. O )
20 eqid 
 |-  ( Base ` K ) = ( Base ` K )
21 20 7 lplnbase 
 |-  ( ( ( G .\/ H ) .\/ I ) e. O -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
22 19 21 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
23 20 2 3 latleeqj1 
 |-  ( ( K e. Lat /\ c e. ( Base ` K ) /\ ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) ) -> ( c .<_ ( ( G .\/ H ) .\/ I ) <-> ( c .\/ ( ( G .\/ H ) .\/ I ) ) = ( ( G .\/ H ) .\/ I ) ) )
24 16 18 22 23 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ ( ( G .\/ H ) .\/ I ) <-> ( c .\/ ( ( G .\/ H ) .\/ I ) ) = ( ( G .\/ H ) .\/ I ) ) )
25 1 2 3 4 5 6 7 8 9 10 dalem28 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P .<_ ( G .\/ c ) )
26 1 dalemkehl 
 |-  ( ph -> K e. HL )
27 26 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
28 5 dalemccea 
 |-  ( ps -> c e. A )
29 28 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
30 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
31 3 4 hlatjcom 
 |-  ( ( K e. HL /\ c e. A /\ G e. A ) -> ( c .\/ G ) = ( G .\/ c ) )
32 27 29 30 31 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ G ) = ( G .\/ c ) )
33 25 32 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P .<_ ( c .\/ G ) )
34 1 2 3 4 5 6 7 8 9 11 dalem33 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Q .<_ ( H .\/ c ) )
35 1 2 3 4 5 6 7 8 9 11 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
36 3 4 hlatjcom 
 |-  ( ( K e. HL /\ c e. A /\ H e. A ) -> ( c .\/ H ) = ( H .\/ c ) )
37 27 29 35 36 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ H ) = ( H .\/ c ) )
38 34 37 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Q .<_ ( c .\/ H ) )
39 1 4 dalempeb 
 |-  ( ph -> P e. ( Base ` K ) )
40 39 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P e. ( Base ` K ) )
41 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ G e. A ) -> ( c .\/ G ) e. ( Base ` K ) )
42 27 29 30 41 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ G ) e. ( Base ` K ) )
43 1 4 dalemqeb 
 |-  ( ph -> Q e. ( Base ` K ) )
44 43 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Q e. ( Base ` K ) )
45 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ H e. A ) -> ( c .\/ H ) e. ( Base ` K ) )
46 27 29 35 45 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ H ) e. ( Base ` K ) )
47 20 2 3 latjlej12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ ( c .\/ G ) e. ( Base ` K ) ) /\ ( Q e. ( Base ` K ) /\ ( c .\/ H ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( c .\/ G ) /\ Q .<_ ( c .\/ H ) ) -> ( P .\/ Q ) .<_ ( ( c .\/ G ) .\/ ( c .\/ H ) ) ) )
48 16 40 42 44 46 47 syl122anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .<_ ( c .\/ G ) /\ Q .<_ ( c .\/ H ) ) -> ( P .\/ Q ) .<_ ( ( c .\/ G ) .\/ ( c .\/ H ) ) ) )
49 33 38 48 mp2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) .<_ ( ( c .\/ G ) .\/ ( c .\/ H ) ) )
50 20 4 atbase 
 |-  ( G e. A -> G e. ( Base ` K ) )
51 30 50 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. ( Base ` K ) )
52 20 4 atbase 
 |-  ( H e. A -> H e. ( Base ` K ) )
53 35 52 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. ( Base ` K ) )
54 20 3 latjjdi 
 |-  ( ( K e. Lat /\ ( c e. ( Base ` K ) /\ G e. ( Base ` K ) /\ H e. ( Base ` K ) ) ) -> ( c .\/ ( G .\/ H ) ) = ( ( c .\/ G ) .\/ ( c .\/ H ) ) )
55 16 18 51 53 54 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ ( G .\/ H ) ) = ( ( c .\/ G ) .\/ ( c .\/ H ) ) )
56 49 55 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) .<_ ( c .\/ ( G .\/ H ) ) )
57 1 2 3 4 5 6 7 8 9 12 dalem37 
 |-  ( ( ph /\ Y = Z /\ ps ) -> R .<_ ( I .\/ c ) )
58 1 2 3 4 5 6 7 8 9 12 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
59 3 4 hlatjcom 
 |-  ( ( K e. HL /\ c e. A /\ I e. A ) -> ( c .\/ I ) = ( I .\/ c ) )
60 27 29 58 59 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ I ) = ( I .\/ c ) )
61 57 60 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> R .<_ ( c .\/ I ) )
62 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
63 62 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) e. ( Base ` K ) )
64 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ G e. A /\ H e. A ) -> ( G .\/ H ) e. ( Base ` K ) )
65 27 30 35 64 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( Base ` K ) )
66 20 3 latjcl 
 |-  ( ( K e. Lat /\ c e. ( Base ` K ) /\ ( G .\/ H ) e. ( Base ` K ) ) -> ( c .\/ ( G .\/ H ) ) e. ( Base ` K ) )
67 16 18 65 66 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ ( G .\/ H ) ) e. ( Base ` K ) )
68 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
69 68 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> R e. ( Base ` K ) )
70 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ I e. A ) -> ( c .\/ I ) e. ( Base ` K ) )
71 27 29 58 70 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ I ) e. ( Base ` K ) )
72 20 2 3 latjlej12 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( c .\/ ( G .\/ H ) ) e. ( Base ` K ) ) /\ ( R e. ( Base ` K ) /\ ( c .\/ I ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .<_ ( c .\/ ( G .\/ H ) ) /\ R .<_ ( c .\/ I ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( c .\/ ( G .\/ H ) ) .\/ ( c .\/ I ) ) ) )
73 16 63 67 69 71 72 syl122anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( P .\/ Q ) .<_ ( c .\/ ( G .\/ H ) ) /\ R .<_ ( c .\/ I ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( c .\/ ( G .\/ H ) ) .\/ ( c .\/ I ) ) ) )
74 56 61 73 mp2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( c .\/ ( G .\/ H ) ) .\/ ( c .\/ I ) ) )
75 20 4 atbase 
 |-  ( I e. A -> I e. ( Base ` K ) )
76 58 75 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. ( Base ` K ) )
77 20 3 latjjdi 
 |-  ( ( K e. Lat /\ ( c e. ( Base ` K ) /\ ( G .\/ H ) e. ( Base ` K ) /\ I e. ( Base ` K ) ) ) -> ( c .\/ ( ( G .\/ H ) .\/ I ) ) = ( ( c .\/ ( G .\/ H ) ) .\/ ( c .\/ I ) ) )
78 16 18 65 76 77 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ ( ( G .\/ H ) .\/ I ) ) = ( ( c .\/ ( G .\/ H ) ) .\/ ( c .\/ I ) ) )
79 74 78 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( c .\/ ( ( G .\/ H ) .\/ I ) ) )
80 8 79 eqbrtrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y .<_ ( c .\/ ( ( G .\/ H ) .\/ I ) ) )
81 breq2 
 |-  ( ( c .\/ ( ( G .\/ H ) .\/ I ) ) = ( ( G .\/ H ) .\/ I ) -> ( Y .<_ ( c .\/ ( ( G .\/ H ) .\/ I ) ) <-> Y .<_ ( ( G .\/ H ) .\/ I ) ) )
82 80 81 syl5ibcom 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ ( ( G .\/ H ) .\/ I ) ) = ( ( G .\/ H ) .\/ I ) -> Y .<_ ( ( G .\/ H ) .\/ I ) ) )
83 24 82 sylbid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ ( ( G .\/ H ) .\/ I ) -> Y .<_ ( ( G .\/ H ) .\/ I ) ) )
84 1 dalemyeo 
 |-  ( ph -> Y e. O )
85 84 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. O )
86 2 7 lplncmp 
 |-  ( ( K e. HL /\ Y e. O /\ ( ( G .\/ H ) .\/ I ) e. O ) -> ( Y .<_ ( ( G .\/ H ) .\/ I ) <-> Y = ( ( G .\/ H ) .\/ I ) ) )
87 27 85 19 86 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( Y .<_ ( ( G .\/ H ) .\/ I ) <-> Y = ( ( G .\/ H ) .\/ I ) ) )
88 83 87 sylibd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ ( ( G .\/ H ) .\/ I ) -> Y = ( ( G .\/ H ) .\/ I ) ) )
89 88 necon3ad 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( Y =/= ( ( G .\/ H ) .\/ I ) -> -. c .<_ ( ( G .\/ H ) .\/ I ) ) )
90 14 89 mpd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ ( ( G .\/ H ) .\/ I ) )