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Metamath Proof Explorer

Theorem dalem54

Description: Lemma for dath . Line G H intersects the auxiliary axis of perspectivity B . (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 dalem54 
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) e. A )

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 dalemkehl 
 |-  ( ph -> K e. HL )
15 14 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
16 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
17 1 2 3 4 5 6 7 8 9 11 dalem29 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. A )
18 1 2 3 4 5 6 7 8 9 10 11 12 dalem41 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G =/= H )
19 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
20 3 4 19 llni2 
 |-  ( ( ( K e. HL /\ G e. A /\ H e. A ) /\ G =/= H ) -> ( G .\/ H ) e. ( LLines ` K ) )
21 15 16 17 18 20 syl31anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( LLines ` K ) )
22 1 2 3 4 5 6 19 7 8 9 10 11 12 13 dalem53 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( LLines ` K ) )
23 1 dalemkelat 
 |-  ( ph -> K e. Lat )
24 23 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
25 eqid 
 |-  ( Base ` K ) = ( Base ` K )
26 25 19 llnbase 
 |-  ( ( G .\/ H ) e. ( LLines ` K ) -> ( G .\/ H ) e. ( Base ` K ) )
27 21 26 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) e. ( Base ` K ) )
28 1 2 3 4 5 6 7 8 9 12 dalem34 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. A )
29 25 4 atbase 
 |-  ( I e. A -> I e. ( Base ` K ) )
30 28 29 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> I e. ( Base ` K ) )
31 25 3 latjcl 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ I e. ( Base ` K ) ) -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
32 24 27 30 31 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) )
33 1 7 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
34 33 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
35 25 2 6 latmle2 
 |-  ( ( K e. Lat /\ ( ( G .\/ H ) .\/ I ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( ( ( G .\/ H ) .\/ I ) ./\ Y ) .<_ Y )
36 24 32 34 35 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( G .\/ H ) .\/ I ) ./\ Y ) .<_ Y )
37 13 36 eqbrtrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B .<_ Y )
38 1 2 3 4 5 6 7 8 9 10 dalem24 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )
39 25 4 atbase 
 |-  ( G e. A -> G e. ( Base ` K ) )
40 16 39 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. ( Base ` K ) )
41 25 4 atbase 
 |-  ( H e. A -> H e. ( Base ` K ) )
42 17 41 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> H e. ( Base ` K ) )
43 25 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( G e. ( Base ` K ) /\ H e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( G .<_ Y /\ H .<_ Y ) <-> ( G .\/ H ) .<_ Y ) )
44 24 40 42 34 43 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .<_ Y /\ H .<_ Y ) <-> ( G .\/ H ) .<_ Y ) )
45 simpl 
 |-  ( ( G .<_ Y /\ H .<_ Y ) -> G .<_ Y )
46 44 45 biimtrrdi 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) .<_ Y -> G .<_ Y ) )
47 38 46 mtod 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. ( G .\/ H ) .<_ Y )
48 nbrne2 
 |-  ( ( B .<_ Y /\ -. ( G .\/ H ) .<_ Y ) -> B =/= ( G .\/ H ) )
49 37 47 48 syl2anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B =/= ( G .\/ H ) )
50 49 necomd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .\/ H ) =/= B )
51 hlatl 
 |-  ( K e. HL -> K e. AtLat )
52 15 51 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. AtLat )
53 25 19 llnbase 
 |-  ( B e. ( LLines ` K ) -> B e. ( Base ` K ) )
54 22 53 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. ( Base ` K ) )
55 25 6 latmcl 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ B e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ B ) e. ( Base ` K ) )
56 24 27 54 55 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) e. ( Base ` K ) )
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 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
59 58 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( P .\/ Q ) e. ( Base ` K ) )
60 25 2 6 latmle1 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) )
61 24 27 59 60 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) )
62 1 2 3 4 5 6 7 8 9 10 11 12 dalem51 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( ( K e. HL /\ c e. A ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) /\ ( ( G .\/ H ) .\/ I ) =/= Y ) )
63 62 simpld 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( K e. HL /\ c e. A ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) )
64 25 4 atbase 
 |-  ( c e. A -> c e. ( Base ` K ) )
65 64 anim2i 
 |-  ( ( K e. HL /\ c e. A ) -> ( K e. HL /\ c e. ( Base ` K ) ) )
66 65 3anim1i 
 |-  ( ( ( K e. HL /\ c e. A ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) )
67 biid 
 |-  ( ( ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) <-> ( ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) )
68 eqid 
 |-  ( ( G .\/ H ) .\/ I ) = ( ( G .\/ H ) .\/ I )
69 eqid 
 |-  ( ( G .\/ H ) ./\ ( P .\/ Q ) ) = ( ( G .\/ H ) ./\ ( P .\/ Q ) )
70 67 2 3 4 6 7 68 8 13 69 dalem10 
 |-  ( ( ( ( K e. HL /\ c e. ( Base ` K ) ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B )
71 66 70 syl3an1 
 |-  ( ( ( ( K e. HL /\ c e. A ) /\ ( G e. A /\ H e. A /\ I e. A ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( ( G .\/ H ) .\/ I ) e. O /\ Y e. O ) /\ ( ( -. c .<_ ( G .\/ H ) /\ -. c .<_ ( H .\/ I ) /\ -. c .<_ ( I .\/ G ) ) /\ ( -. c .<_ ( P .\/ Q ) /\ -. c .<_ ( Q .\/ R ) /\ -. c .<_ ( R .\/ P ) ) /\ ( c .<_ ( G .\/ P ) /\ c .<_ ( H .\/ Q ) /\ c .<_ ( I .\/ R ) ) ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B )
72 63 71 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B )
73 25 6 latmcl 
 |-  ( ( K e. Lat /\ ( G .\/ H ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. ( Base ` K ) )
74 24 27 59 73 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. ( Base ` K ) )
75 25 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 ) ) )
76 24 74 27 54 75 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( G .\/ H ) /\ ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ B ) <-> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) ) )
77 61 72 76 mpbi2and 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) )
78 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
79 25 2 78 4 atlen0 
 |-  ( ( ( K e. AtLat /\ ( ( G .\/ H ) ./\ B ) e. ( Base ` K ) /\ ( ( G .\/ H ) ./\ ( P .\/ Q ) ) e. A ) /\ ( ( G .\/ H ) ./\ ( P .\/ Q ) ) .<_ ( ( G .\/ H ) ./\ B ) ) -> ( ( G .\/ H ) ./\ B ) =/= ( 0. ` K ) )
80 52 56 57 77 79 syl31anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) =/= ( 0. ` K ) )
81 6 78 4 19 2llnmat 
 |-  ( ( ( K e. HL /\ ( G .\/ H ) e. ( LLines ` K ) /\ B e. ( LLines ` K ) ) /\ ( ( G .\/ H ) =/= B /\ ( ( G .\/ H ) ./\ B ) =/= ( 0. ` K ) ) ) -> ( ( G .\/ H ) ./\ B ) e. A )
82 15 21 22 50 80 81 syl32anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( G .\/ H ) ./\ B ) e. A )