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Metamath Proof Explorer

Theorem dalem60

Description: Lemma for dath . B is an axis of perspectivity (almost). (Contributed by NM, 11-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 ) ) ) )
dalem60.m 
|- ./\ = ( meet ` K )
dalem60.o 
|- O = ( LPlanes ` K )
dalem60.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem60.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem60.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem60.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
dalem60.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem60.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem60.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem60.b1 
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem60 
|- ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) = 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 dalem60.m 
 |-  ./\ = ( meet ` K )
7 dalem60.o 
 |-  O = ( LPlanes ` K )
8 dalem60.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem60.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem60.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
11 dalem60.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
12 dalem60.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
13 dalem60.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
14 dalem60.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
15 dalem60.b1 
 |-  B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
16 1 2 3 4 5 6 7 8 9 10 12 13 14 15 dalem57 
 |-  ( ( ph /\ Y = Z /\ ps ) -> D .<_ B )
17 1 2 3 4 5 6 7 8 9 11 12 13 14 15 dalem58 
 |-  ( ( ph /\ Y = Z /\ ps ) -> E .<_ B )
18 1 dalemkelat 
 |-  ( ph -> K e. Lat )
19 18 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
20 1 2 3 4 6 7 8 9 10 dalemdea 
 |-  ( ph -> D e. A )
21 eqid 
 |-  ( Base ` K ) = ( Base ` K )
22 21 4 atbase 
 |-  ( D e. A -> D e. ( Base ` K ) )
23 20 22 syl 
 |-  ( ph -> D e. ( Base ` K ) )
24 23 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> D e. ( Base ` K ) )
25 1 2 3 4 6 7 8 9 11 dalemeea 
 |-  ( ph -> E e. A )
26 21 4 atbase 
 |-  ( E e. A -> E e. ( Base ` K ) )
27 25 26 syl 
 |-  ( ph -> E e. ( Base ` K ) )
28 27 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> E e. ( Base ` K ) )
29 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
30 1 2 3 4 5 6 29 7 8 9 12 13 14 15 dalem53 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( LLines ` K ) )
31 21 29 llnbase 
 |-  ( B e. ( LLines ` K ) -> B e. ( Base ` K ) )
32 30 31 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( Base ` K ) )
33 21 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( D e. ( Base ` K ) /\ E e. ( Base ` K ) /\ B e. ( Base ` K ) ) ) -> ( ( D .<_ B /\ E .<_ B ) <-> ( D .\/ E ) .<_ B ) )
34 19 24 28 32 33 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( D .<_ B /\ E .<_ B ) <-> ( D .\/ E ) .<_ B ) )
35 16 17 34 mpbi2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) .<_ B )
36 1 dalemkehl 
 |-  ( ph -> K e. HL )
37 36 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
38 1 2 3 4 6 7 8 9 10 11 dalemdnee 
 |-  ( ph -> D =/= E )
39 3 4 29 llni2 
 |-  ( ( ( K e. HL /\ D e. A /\ E e. A ) /\ D =/= E ) -> ( D .\/ E ) e. ( LLines ` K ) )
40 36 20 25 38 39 syl31anc 
 |-  ( ph -> ( D .\/ E ) e. ( LLines ` K ) )
41 40 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) e. ( LLines ` K ) )
42 2 29 llncmp 
 |-  ( ( K e. HL /\ ( D .\/ E ) e. ( LLines ` K ) /\ B e. ( LLines ` K ) ) -> ( ( D .\/ E ) .<_ B <-> ( D .\/ E ) = B ) )
43 37 41 30 42 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( D .\/ E ) .<_ B <-> ( D .\/ E ) = B ) )
44 35 43 mpbid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) = B )