This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dalem53

Description: Lemma for dath . The auxiliary axis of perspectivity B is a line (analogous to the actual axis of perspectivity X in dalem15 . (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 ) ) ) )
dalem53.m 
|- ./\ = ( meet ` K )
dalem53.n 
|- N = ( LLines ` K )
dalem53.o 
|- O = ( LPlanes ` K )
dalem53.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem53.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem53.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem53.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem53.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
dalem53.b1 
|- B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
Assertion dalem53 
|- ( ( ph /\ Y = Z /\ ps ) -> B e. N )

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 dalem53.m 
 |-  ./\ = ( meet ` K )
7 dalem53.n 
 |-  N = ( LLines ` K )
8 dalem53.o 
 |-  O = ( LPlanes ` K )
9 dalem53.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
10 dalem53.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
11 dalem53.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
12 dalem53.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
13 dalem53.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
14 dalem53.b1 
 |-  B = ( ( ( G .\/ H ) .\/ I ) ./\ Y )
15 1 2 3 4 5 6 8 9 10 11 12 13 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 ) )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 4 atbase 
 |-  ( c e. A -> c e. ( Base ` K ) )
18 17 anim2i 
 |-  ( ( K e. HL /\ c e. A ) -> ( K e. HL /\ c e. ( Base ` K ) ) )
19 18 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 ) ) )
20 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 ) ) ) ) )
21 eqid 
 |-  ( ( G .\/ H ) .\/ I ) = ( ( G .\/ H ) .\/ I )
22 20 2 3 4 6 7 8 21 9 14 dalem15 
 |-  ( ( ( ( ( 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 ) .\/ I ) =/= Y ) -> B e. N )
23 19 22 syl3anl1 
 |-  ( ( ( ( ( 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 ) -> B e. N )
24 15 23 syl 
 |-  ( ( ph /\ Y = Z /\ ps ) -> B e. N )