Metamath Proof Explorer


Theorem dalem57

Description: Lemma for dath . Axis of perspectivity point D is on the auxiliary line B . (Contributed by NM, 9-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 ) ) ) )
dalem57.m
|- ./\ = ( meet ` K )
dalem57.o
|- O = ( LPlanes ` K )
dalem57.y
|- Y = ( ( P .\/ Q ) .\/ R )
dalem57.z
|- Z = ( ( S .\/ T ) .\/ U )
dalem57.d
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem57.g
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem57.h
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem57.i
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem57.b1
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem57
|- ( ( ph /\ Y = Z /\ ps ) -> D .<_ B )

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 dalem57.m
 |-  ./\ = ( meet ` K )
7 dalem57.o
 |-  O = ( LPlanes ` K )
8 dalem57.y
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem57.z
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem57.d
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
11 dalem57.g
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
12 dalem57.h
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
13 dalem57.i
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
14 dalem57.b1
 |-  B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
15 1 2 3 4 5 6 7 8 9 11 12 13 14 dalem55
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ B ) )
16 1 dalemkelat
 |-  ( ph -> K e. Lat )
17 16 3ad2ant1
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
18 1 dalemkehl
 |-  ( ph -> K e. HL )
19 18 3ad2ant1
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
20 1 2 3 4 5 6 7 8 9 11 dalem23
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
21 1 2 3 4 5 6 7 8 9 12 dalem29
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
22 eqid
 |-  ( Base ` K ) = ( Base ` K )
23 22 3 4 hlatjcl
 |-  ( ( K e. HL /\ G e. A /\ H e. A ) -> ( G .\/ H ) e. ( Base ` K ) )
24 19 20 21 23 syl3anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( Base ` K ) )
25 1 3 4 dalempjqeb
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
26 25 3ad2ant1
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) e. ( Base ` K ) )
27 22 2 6 latmle2
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( P .\/ Q ) )
28 17 24 26 27 syl3anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( P .\/ Q ) )
29 15 28 eqbrtrrd
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) .<_ ( P .\/ Q ) )
30 1 2 3 4 5 6 7 8 9 11 12 13 14 dalem56
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( S .\/ T ) ) = ( ( G .\/ H ) ./\ B ) )
31 1 3 4 dalemsjteb
 |-  ( ph -> ( S .\/ T ) e. ( Base ` K ) )
32 31 3ad2ant1
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( S .\/ T ) e. ( Base ` K ) )
33 22 2 6 latmle2
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ ( S .\/ T ) ) .<_ ( S .\/ T ) )
34 17 24 32 33 syl3anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( S .\/ T ) ) .<_ ( S .\/ T ) )
35 30 34 eqbrtrrd
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) .<_ ( S .\/ T ) )
36 1 2 3 4 5 6 7 8 9 11 12 13 14 dalem54
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) e. A )
37 22 4 atbase
 |-  ( ( ( G .\/ H ) ./\ B ) e. A -> ( ( G .\/ H ) ./\ B ) e. ( Base ` K ) )
38 36 37 syl
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) e. ( Base ` K ) )
39 22 2 6 latlem12
 |-  ( ( K e. Lat /\ ( ( ( G .\/ H ) ./\ B ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) ) -> ( ( ( ( G .\/ H ) ./\ B ) .<_ ( P .\/ Q ) /\ ( ( G .\/ H ) ./\ B ) .<_ ( S .\/ T ) ) <-> ( ( G .\/ H ) ./\ B ) .<_ ( ( P .\/ Q ) ./\ ( S .\/ T ) ) ) )
40 17 38 26 32 39 syl13anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( ( G .\/ H ) ./\ B ) .<_ ( P .\/ Q ) /\ ( ( G .\/ H ) ./\ B ) .<_ ( S .\/ T ) ) <-> ( ( G .\/ H ) ./\ B ) .<_ ( ( P .\/ Q ) ./\ ( S .\/ T ) ) ) )
41 29 35 40 mpbi2and
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) .<_ ( ( P .\/ Q ) ./\ ( S .\/ T ) ) )
42 41 10 breqtrrdi
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) .<_ D )
43 hlatl
 |-  ( K e. HL -> K e. AtLat )
44 19 43 syl
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
45 1 2 3 4 6 7 8 9 10 dalemdea
 |-  ( ph -> D e. A )
46 45 3ad2ant1
 |-  ( ( ph /\ Y = Z /\ ps ) -> D e. A )
47 2 4 atcmp
 |-  ( ( K e. AtLat /\ ( ( G .\/ H ) ./\ B ) e. A /\ D e. A ) -> ( ( ( G .\/ H ) ./\ B ) .<_ D <-> ( ( G .\/ H ) ./\ B ) = D ) )
48 44 36 46 47 syl3anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) ./\ B ) .<_ D <-> ( ( G .\/ H ) ./\ B ) = D ) )
49 42 48 mpbid
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) = D )
50 eqid
 |-  ( LLines ` K ) = ( LLines ` K )
51 1 2 3 4 5 6 50 7 8 9 11 12 13 14 dalem53
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( LLines ` K ) )
52 22 50 llnbase
 |-  ( B e. ( LLines ` K ) -> B e. ( Base ` K ) )
53 51 52 syl
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( Base ` K ) )
54 22 2 6 latmle2
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ B e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ B ) .<_ B )
55 17 24 53 54 syl3anc
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) .<_ B )
56 49 55 eqbrtrrd
 |-  ( ( ph /\ Y = Z /\ ps ) -> D .<_ B )