Metamath Proof Explorer

Theorem dalem55

Description: Lemma for dath . Lines G H and P Q intersect at the auxiliary line B (later shown to be an axis of perspectivity; see dalem60 ). (Contributed by NM, 8-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 ) ) ) )
dalem54.m 
|- ./\ = ( meet ` K )
dalem54.o 
|- O = ( LPlanes ` K )
dalem54.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem54.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem54.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem54.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem54.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem54.b1 
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem55 
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ 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 dalem54.m 
 |-  ./\ = ( meet ` K )
7 dalem54.o 
 |-  O = ( LPlanes ` K )
8 dalem54.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem54.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem54.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 dalem54.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
12 dalem54.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
13 dalem54.b1 
 |-  B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
14 1 dalemkelat 
 |-  ( ph -> K e. Lat )
15 14 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
16 1 dalemkehl 
 |-  ( ph -> K e. HL )
17 16 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
18 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
19 1 2 3 4 5 6 7 8 9 11 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
20 eqid 
 |-  ( Base ` K ) = ( Base ` K )
21 20 3 4 hlatjcl 
 |-  ( ( K e. HL /\ G e. A /\ H e. A ) -> ( G .\/ H ) e. ( Base ` K ) )
22 17 18 19 21 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( Base ` K ) )
23 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
24 23 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) e. ( Base ` K ) )
25 20 2 6 latmle1 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) )
26 15 22 24 25 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) )
27 1 2 3 4 5 6 7 8 9 12 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
28 20 4 atbase 
 |-  ( I e. A -> I e. ( Base ` K ) )
29 27 28 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. ( Base ` K ) )
30 20 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ I e. ( Base ` K ) ) -> ( G .\/ H ) .<_ ( ( G .\/ H ) .\/ I ) )
31 15 22 29 30 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) .<_ ( ( G .\/ H ) .\/ I ) )
32 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
33 20 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ R ) )
34 14 23 32 33 syl3anc 
 |-  ( ph -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ R ) )
35 34 8 breqtrrdi 
 |-  ( ph -> ( P .\/ Q ) .<_ Y )
36 35 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) .<_ Y )
37 1 2 3 4 5 6 7 8 9 10 11 12 dalem42 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. O )
38 20 7 lplnbase 
 |-  ( ( ( G .\/ H ) .\/ I ) e. O -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
39 37 38 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
40 1 7 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
41 40 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
42 20 2 6 latmlem12 
 |-  ( ( K e. Lat /\ ( ( G .\/ H ) e. ( Base ` K ) /\ ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) ) /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( ( G .\/ H ) .<_ ( ( G .\/ H ) .\/ I ) /\ ( P .\/ Q ) .<_ Y ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( ( G .\/ H ) .\/ I ) ./\ Y ) ) )
43 15 22 39 24 41 42 syl122anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) .<_ ( ( G .\/ H ) .\/ I ) /\ ( P .\/ Q ) .<_ Y ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( ( G .\/ H ) .\/ I ) ./\ Y ) ) )
44 31 36 43 mp2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( ( G .\/ H ) .\/ I ) ./\ Y ) )
45 44 13 breqtrrdi 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B )
46 20 6 latmcl 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. ( Base ` K ) )
47 15 22 24 46 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. ( Base ` K ) )
48 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
49 1 2 3 4 5 6 48 7 8 9 10 11 12 13 dalem53 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( LLines ` K ) )
50 20 48 llnbase 
 |-  ( B e. ( LLines ` K ) -> B e. ( Base ` K ) )
51 49 50 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( Base ` K ) )
52 20 2 6 latlem12 
 |-  ( ( K e. Lat /\ ( ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. ( Base ` K ) /\ ( G .\/ H ) e. ( Base ` K ) /\ B e. ( Base ` K ) ) ) -> ( ( ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) /\ ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B ) <-> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) ) )
53 15 47 22 51 52 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) /\ ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B ) <-> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) ) )
54 26 45 53 mpbi2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) )
55 hlatl 
 |-  ( K e. HL -> K e. AtLat )
56 17 55 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
57 1 2 3 4 5 6 7 8 9 10 11 12 dalem52 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. A )
58 1 2 3 4 5 6 7 8 9 10 11 12 13 dalem54 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) e. A )
59 2 4 atcmp 
 |-  ( ( K e. AtLat /\ ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. A /\ ( ( G .\/ H ) ./\ B ) e. A ) -> ( ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) <-> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ B ) ) )
60 56 57 58 59 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) <-> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ B ) ) )
61 54 60 mpbid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ B ) )