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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 )