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Theorem List for Metamath Proof Explorer - 34501-34600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdalem33 34501 Lemma for dath 34532. Analog of dalem28 34496 for  H. (Contributed by NM, 4-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( H  .\/  c )
 )
 
Theoremdalem34 34502 Lemma for dath 34532. Analog of dalem23 34492 for  I. (Contributed by NM, 2-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
 
Theoremdalem35 34503 Lemma for dath 34532. Analog of dalem24 34493 for  I. (Contributed by NM, 3-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  I  .<_  Y )
 
Theoremdalem36 34504 Lemma for dath 34532. Analog of dalem27 34495 for  I. (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  c  .<_  ( I  .\/  R )
 )
 
Theoremdalem37 34505 Lemma for dath 34532. Analog of dalem28 34496 for  I. (Contributed by NM, 4-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( I  .\/  c )
 )
 
Theoremdalem38 34506 Lemma for dath 34532. Plane  Y belongs to the 3-dimensional volume  G H I c. (Contributed by NM, 5-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H )  .\/  I )  .\/  c ) )
 
Theoremdalem39 34507 Lemma for dath 34532. Auxiliary atoms  G,  H, and  I are not colinear. (Contributed by NM, 4-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  H  .<_  ( I  .\/  G )
 )
 
Theoremdalem40 34508 Lemma for dath 34532. Analog of dalem39 34507 for  I. (Contributed by NM, 4-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  I  .<_  ( G  .\/  H )
 )
 
Theoremdalem41 34509 Lemma for dath 34532. (Contributed by NM, 4-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  H )
 
Theoremdalem42 34510 Lemma for dath 34532. Auxiliary atoms  G H I form a plane. (Contributed by NM, 4-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( ( G  .\/  H )  .\/  I )  e.  O )
 
Theoremdalem43 34511 Lemma for dath 34532. Planes  G H I and  Y are different. (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( ( G  .\/  H )  .\/  I )  =/=  Y )
 
Theoremdalem44 34512 Lemma for dath 34532. Dummy center of perspectivity  c lies outside of plane  G H I. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( ( G  .\/  H )  .\/  I )
 )
 
Theoremdalem45 34513 Lemma for dath 34532. Dummy center of perspectivity  c is not on the line  G H. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( G  .\/  H ) )
 
Theoremdalem46 34514 Lemma for dath 34532. Analog of dalem45 34513 for  H I. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( H  .\/  I
 ) )
 
Theoremdalem47 34515 Lemma for dath 34532. Analog of dalem45 34513 for  I G. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( I  .\/  G ) )
 
Theoremdalem48 34516 Lemma for dath 34532. Analog of dalem45 34513 for  P Q. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  -.  c  .<_  ( P  .\/  Q ) )
 
Theoremdalem49 34517 Lemma for dath 34532. Analog of dalem45 34513 for  Q R. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  -.  c  .<_  ( Q  .\/  R ) )
 
Theoremdalem50 34518 Lemma for dath 34532. Analog of dalem45 34513 for  R P. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  -.  c  .<_  ( R  .\/  P ) )
 
Theoremdalem51 34519 Lemma for dath 34532. Construct the condition  ph with  c,  G H I, and 
Y in place of  C,  Y, and  Z respectively. This lets us reuse the special case of Desargues' Theorem where  Y  =/=  Z, to eventually prove the case where  Y  =  Z. (Contributed by NM, 16-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( 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 )
 )
 
Theoremdalem52 34520 Lemma for dath 34532. Lines  G H and  P Q intersect at an atom. (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
 
Theoremdalem53 34521 Lemma for dath 34532. The auxliary axis of perspectivity  B is a line (analogous to the actual axis of perspectivity  X in dalem15 34474. (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  N  =  ( LLines `  K )   &    |-  O  =  (
 LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  B  e.  N )
 
Theoremdalem54 34522 Lemma for dath 34532. Line  G H intersects the auxiliary axis of perspectivity  B. (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( ( G  .\/  H )  ./\  B )  e.  A )
 
Theoremdalem55 34523 Lemma for dath 34532. Lines  G H and  P Q intersect at the auxiliary line  B (later shown to be an axis of perspectivity; see dalem60 34528). (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\ 
 B ) )
 
Theoremdalem56 34524 Lemma for dath 34532. Analog of dalem55 34523 for line  S T. (Contributed by NM, 8-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\ 
 B ) )
 
Theoremdalem57 34525 Lemma for dath 34532. Axis of perspectivity point  D is on the auxiliary line  B. (Contributed by NM, 9-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
 
Theoremdalem58 34526 Lemma for dath 34532. Analog of dalem57 34525 for  E. (Contributed by NM, 10-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )
 
Theoremdalem59 34527 Lemma for dath 34532. Analog of dalem57 34525 for  F. (Contributed by NM, 10-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  B )
 
Theoremdalem60 34528 Lemma for dath 34532. 
B is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )   &    |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )   &    |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )   &    |-  B  =  ( ( ( G 
 .\/  H )  .\/  I
 )  ./\  Y )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  ( D  .\/  E )  =  B )
 
Theoremdalem61 34529 Lemma for dath 34532. Show that atoms  D,  E, and  F lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms  c and  d. (Contributed by NM, 11-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ( ps 
 <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
 d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
 ) ) ) )   &    |-  ./\ 
 =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )   =>    |-  ( ( ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( D  .\/  E )
 )
 
Theoremdalem62 34530 Lemma for dath 34532. Eliminate the condition  ps containing dummy variables  c and  d. (Contributed by NM, 11-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )   =>    |-  ( ( ph  /\  Y  =  Z ) 
 ->  F  .<_  ( D  .\/  E ) )
 
Theoremdalem63 34531 Lemma for dath 34532. Combine the cases where  Y and  Z are different planes with the case where  Y and 
Z are the same plane. (Contributed by NM, 11-Aug-2012.)
 |-  ( 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 ) ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  Y  =  ( ( P  .\/  Q )  .\/  R )   &    |-  Z  =  ( ( S  .\/  T )  .\/  U )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )   =>    |-  ( ph  ->  F 
 .<_  ( D  .\/  E ) )
 
Theoremdath 34532 Desargues' Theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points)  P Q R and  S T U forms a triangle (i.e. determines a plane). Assume that lines  P S,  Q T, and  R U meet at a "center of perspectivity"  C. (We also assume that  C is not on any of the 6 lines forming the two triangles.) Then the atoms 
D  =  ( P 
.\/  Q )  ./\  ( S  .\/  T ),  E  =  ( Q  .\/  R ) 
./\  ( T  .\/  U ),  F  =  ( R  .\/  P ) 
./\  ( U  .\/  S ) are colinear, forming an "axis of perspectivity".

Our proof roughly follows Theorem 2.7.1, p. 78 in Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press (1988). Unlike them, we don't assume  C is an atom to make this theorem slightly more general for easier future use. However, we prove that 
C must be an atom in dalemcea 34456.

For a visual demonstration, see the "Desargue's Theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html. The points I, J, and K there define the axis of perspectivity.

See theorem dalaw 34682 for Desargues Law, which eliminates all of the preconditions on the atoms except for central perspectivity. This is Metamath 100 proof #87. (Contributed by NM, 20-Aug-2012.)

 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )   =>    |-  ( ( ( ( K  e.  HL  /\  C  e.  B ) 
 /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( ( S  .\/  T )  .\/  U )  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 ) ) ) ) 
 ->  F  .<_  ( D  .\/  E ) )
 
Theoremdath2 34533 Version of Desargues' Theorem dath 34532 with a different variable ordering. (Contributed by NM, 7-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  O  =  ( LPlanes `  K )   &    |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )   &    |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )   &    |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )   =>    |-  ( ( ( ( K  e.  HL  /\  C  e.  B ) 
 /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( ( ( P  .\/  Q )  .\/  R )  e.  O  /\  ( ( S  .\/  T )  .\/  U )  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 ) ) ) ) 
 ->  D  .<_  ( E  .\/  F ) )
 
Theoremlineset 34534* The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   =>    |-  ( K  e.  B  ->  N  =  { s  | 
 E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } ) } )
 
Theoremisline 34535* The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
 
Theoremislinei 34536* Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   =>    |-  (
 ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A ) 
 /\  ( Q  =/=  R 
 /\  X  =  { p  e.  A  |  p  .<_  ( Q  .\/  R ) } ) ) 
 ->  X  e.  N )
 
TheorempointsetN 34537* The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( Points `  K )   =>    |-  ( K  e.  B  ->  P  =  { p  |  E. a  e.  A  p  =  { a } } )
 
TheoremispointN 34538* The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( Points `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  P  <->  E. a  e.  A  X  =  { a } ) )
 
TheorematpointN 34539 The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( Points `  K )   =>    |-  (
 ( K  e.  D  /\  X  e.  A ) 
 ->  { X }  e.  P )
 
Theorempsubspset 34540* The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  ( K  e.  B  ->  S  =  { s  |  ( s  C_  A  /\  A. p  e.  s  A. q  e.  s  A. r  e.  A  ( r  .<_  ( p 
 .\/  q )  ->  r  e.  s )
 ) } )
 
Theoremispsubsp 34541* The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  S  <->  ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r  .<_  ( p  .\/  q )  ->  r  e.  X ) ) ) )
 
Theoremispsubsp2 34542* The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  S  <->  ( X  C_  A  /\  A. p  e.  A  ( E. q  e.  X  E. r  e.  X  p  .<_  ( q 
 .\/  r )  ->  p  e.  X )
 ) ) )
 
Theorempsubspi 34543* Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A ) 
 /\  E. q  e.  X  E. r  e.  X  P  .<_  ( q  .\/  r ) )  ->  P  e.  X )
 
Theorempsubspi2N 34544 Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( ( K  e.  D  /\  X  e.  S  /\  P  e.  A ) 
 /\  ( Q  e.  X  /\  R  e.  X  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  e.  X )
 
Theorem0psubN 34545 The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   =>    |-  ( K  e.  V  ->  (/)  e.  S )
 
TheoremsnatpsubN 34546 The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( K  e.  AtLat  /\  P  e.  A ) 
 ->  { P }  e.  S )
 
TheorempointpsubN 34547 A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
 |-  P  =  ( Points `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( K  e.  AtLat  /\  X  e.  P ) 
 ->  X  e.  S )
 
TheoremlinepsubN 34548 A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
 |-  N  =  ( Lines `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  N ) 
 ->  X  e.  S )
 
TheorematpsubN 34549 The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  ( K  e.  V  ->  A  e.  S )
 
Theorempsubssat 34550 A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( K  e.  B  /\  X  e.  S ) 
 ->  X  C_  A )
 
TheorempsubatN 34551 A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   =>    |-  (
 ( K  e.  B  /\  X  e.  S  /\  Y  e.  X )  ->  Y  e.  A )
 
Theorempmapfval 34552* The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } )
 )
 
Theorempmapval 34553* Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X )  =  { a  e.  A  |  a  .<_  X }
 )
 
Theoremelpmap 34554 Member of a projective map. (Contributed by NM, 27-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( P  e.  ( M `  X )  <->  ( P  e.  A  /\  P  .<_  X ) ) )
 
Theorempmapssat 34555 The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  C  /\  X  e.  B ) 
 ->  ( M `  X )  C_  A )
 
TheorempmapssbaN 34556 A weakening of pmapssat 34555 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  C  /\  X  e.  B ) 
 ->  ( M `  X )  C_  B )
 
Theorempmaple 34557 The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  <->  ( M `  X )  C_  ( M `
  Y ) ) )
 
Theorempmap11 34558 The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( M `  X )  =  ( M `  Y )  <->  X  =  Y ) )
 
Theorempmapat 34559 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A ) 
 ->  ( M `  P )  =  { P } )
 
Theoremelpmapat 34560 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)
 |-  A  =  ( Atoms `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A ) 
 ->  ( X  e.  ( M `  P )  <->  X  =  P ) )
 
Theorempmap0 34561 Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
 |-  .0.  =  ( 0. `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
 
Theorempmapeq0 34562 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `
  X )  =  (/) 
 <->  X  =  .0.  )
 )
 
Theorempmap1N 34563 Value of the projective map of a Hilbert lattice at lattice unit. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)
 |-  .1.  =  ( 1. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( K  e.  OP  ->  ( M `  .1.  )  =  A )
 
Theorempmapsub 34564 The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( M `  X )  e.  S )
 
Theorempmapglbx 34565* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 34566, where we read  S as  S ( i ). Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B  /\  I  =/=  (/) )  ->  ( M `  ( G `
  { y  | 
 E. i  e.  I  y  =  S }
 ) )  =  |^|_ i  e.  I  ( M `
  S ) )
 
Theorempmapglb 34566* The projective map of the GLB of a set of lattice elements  S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `
  S ) )  =  |^|_ x  e.  S  ( M `  x ) )
 
Theorempmapglb2N 34567* The projective map of the GLB of a set of lattice elements  S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows  S  =  (/). (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  HL  /\  S  C_  B )  ->  ( M `  ( G `  S ) )  =  ( A  i^i  |^|_ x  e.  S  ( M `  x ) ) )
 
Theorempmapglb2xN 34568* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 34567, where we read  S as  S ( i ). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows  I  =  (/). (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( K  e.  HL  /\  A. i  e.  I  S  e.  B )  ->  ( M `  ( G `  { y  |  E. i  e.  I  y  =  S } ) )  =  ( A  i^i  |^|_ i  e.  I  ( M `  S ) ) )
 
Theorempmapmeet 34569 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( ( P `
  X )  i^i  ( P `  Y ) ) )
 
Theoremisline2 34570* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  ( K  e.  Lat  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( M `
  ( p  .\/  q ) ) ) ) )
 
Theoremlinepmap 34571 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q ) 
 ->  ( M `  ( P  .\/  Q ) )  e.  N )
 
Theoremisline3 34572* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q
 ) ) ) )
 
Theoremisline4N 34573* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  p C X ) )
 
Theoremlneq2at 34574 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( P  .<_  X  /\  Q  .<_  X ) ) 
 ->  X  =  ( P 
 .\/  Q ) )
 
TheoremlnatexN 34575* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  ( M `  X )  e.  N )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<_  X ) )
 
TheoremlnjatN 34576* Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A ) 
 /\  ( ( M `
  X )  e.  N  /\  P  .<_  X ) )  ->  E. q  e.  A  ( q  =/= 
 P  /\  X  =  ( P  .\/  q ) ) )
 
TheoremlncvrelatN 34577 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B ) 
 /\  ( ( M `
  X )  e.  N  /\  P C X ) )  ->  P  e.  A )
 
Theoremlncvrat 34578 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A ) 
 /\  ( ( M `
  X )  e.  N  /\  P  .<_  X ) )  ->  P C X )
 
Theoremlncmp 34579 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  N  =  ( Lines `  K )   &    |-  M  =  (
 pmap `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( ( M `  X )  e.  N  /\  ( M `  Y )  e.  N )
 )  ->  ( X  .<_  Y  <->  X  =  Y ) )
 
Theorem2lnat 34580 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 Lines `  K )   &    |-  F  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( ( F `
  X )  e.  N  /\  ( F `
  Y )  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y )  =/= 
 .0.  ) )  ->  ( X  ./\  Y )  e.  A )
 
Theorem2atm2atN 34581 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =/=  .0.  )
 
Theorem2llnma1b 34582 Generalization of 2llnma1 34583. (Contributed by NM, 26-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )  /\  -.  Q  .<_  ( P 
 .\/  X ) )  ->  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) )  =  P )
 
Theorem2llnma1 34583 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  -.  R  .<_  ( P 
 .\/  Q ) )  ->  ( ( Q  .\/  P )  ./\  ( Q  .\/  R ) )  =  Q )
 
Theorem2llnma3r 34584 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .\/  R )  =/=  ( Q  .\/  R ) )  ->  (
 ( P  .\/  R )  ./\  ( Q  .\/  R ) )  =  R )
 
Theorem2llnma2 34585 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  R )
 
Theorem2llnma2rN 34586 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( ( P  .\/  R )  ./\  ( Q  .\/  R ) )  =  R )
 
21.30.10  Construction of a vector space from a Hilbert lattice
 
Theoremcdlema1N 34587 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  ( Lines `  K )   &    |-  F  =  ( pmap `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  ( ( R  =/=  P  /\  R  .<_  ( P  .\/  Q ) )  /\  ( P 
 .<_  X  /\  Q  .<_  Y )  /\  ( ( F `  Y )  e.  N  /\  ( X  ./\  Y )  e.  A  /\  -.  Q  .<_  X ) ) ) 
 ->  ( X  .\/  R )  =  ( X  .\/  Y ) )
 
Theoremcdlema2N 34588 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( ( R  =/=  P 
 /\  R  .<_  ( P 
 .\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  ( R  ./\ 
 X )  =  .0.  )
 
Theoremcdlemblem 34589 Lemma for cdlemb 34590. (Contributed by NM, 8-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .<  =  ( lt `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  V  =  ( ( P  .\/  Q )  ./\ 
 X )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q ) 
 /\  ( X C  .1.  /\  -.  P  .<_  X 
 /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  ( u  =/=  V  /\  u  .<  X ) ) 
 /\  ( r  e.  A  /\  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u )
 ) ) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Theoremcdlemb 34590* Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q ) 
 /\  ( X C  .1.  /\  -.  P  .<_  X 
 /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Syntaxcpadd 34591 Extend class notation with projective subspace sum.
 class  +P
 
Definitiondf-padd 34592* Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
 |-  +P  =  ( l  e.  _V  |->  ( m  e. 
 ~P ( Atoms `  l
 ) ,  n  e. 
 ~P ( Atoms `  l
 )  |->  ( ( m  u.  n )  u. 
 { p  e.  ( Atoms `  l )  | 
 E. q  e.  m  E. r  e.  n  p ( le `  l
 ) ( q (
 join `  l ) r ) } ) ) )
 
Theorempaddfval 34593* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( K  e.  B  ->  .+  =  ( m  e.  ~P A ,  n  e.  ~P A  |->  ( ( m  u.  n )  u.  { p  e.  A  |  E. q  e.  m  E. r  e.  n  p  .<_  ( q  .\/  r ) } ) ) )
 
Theorempaddval 34594* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( ( X  u.  Y )  u. 
 { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r ) } )
 )
 
Theoremelpadd 34595* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  ( S  e.  ( X  .+  Y )  <->  ( ( S  e.  X  \/  S  e.  Y )  \/  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S  .<_  ( q  .\/  r
 ) ) ) ) )
 
Theoremelpaddn0 34596* Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) ) 
 ->  ( S  e.  ( X  .+  Y )  <->  ( S  e.  A  /\  E. q  e.  X  E. r  e.  Y  S  .<_  ( q 
 .\/  r ) ) ) )
 
Theorempaddvaln0N 34597* Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) ) 
 ->  ( X  .+  Y )  =  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p  .<_  ( q 
 .\/  r ) }
 )
 
Theoremelpaddri 34598 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( Q  e.  X  /\  R  e.  Y )  /\  ( S  e.  A  /\  S  .<_  ( Q 
 .\/  R ) ) ) 
 ->  S  e.  ( X 
 .+  Y ) )
 
TheoremelpaddatriN 34599 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  ( R  e.  X  /\  S  e.  A  /\  S  .<_  ( R  .\/  Q ) ) )  ->  S  e.  ( X  .+ 
 { Q } )
 )
 
Theoremelpaddat 34600* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  .+  =  ( +P `  K )   =>    |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Q  e.  A )  /\  X  =/=  (/) )  ->  ( S  e.  ( X  .+  { Q }
 ) 
 <->  ( S  e.  A  /\  E. p  e.  X  S  .<_  ( p  .\/  Q ) ) ) )
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