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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtendoplcom 30101* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( V P U ) )
 
Theoremtendoplass 30102* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  U  e.  E  /\  V  e.  E )
 )  ->  ( ( S P U ) P V )  =  ( S P ( U P V ) ) )
 
Theoremtendodi1 30103* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  U  e.  E  /\  V  e.  E )
 )  ->  ( S  o.  ( U P V ) )  =  (
 ( S  o.  U ) P ( S  o.  V ) ) )
 
Theoremtendodi2 30104* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  U  e.  E  /\  V  e.  E )
 )  ->  ( ( S P U )  o.  V )  =  ( ( S  o.  V ) P ( U  o.  V ) ) )
 
Theoremtendo0cbv 30105* Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)
 |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
 
Theoremtendo02 30106* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
 |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  B  =  ( Base `  K )   =>    |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
 
Theoremtendo0co2 30107* The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 30340? (Contributed by NM, 11-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( O `
  ( F  o.  G ) )  =  ( ( O `  F )  o.  ( O `  G ) ) )
 
Theoremtendo0tp 30108* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |- 
 .<_  =  ( le `  K )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  ( R `  ( O `  F ) )  .<_  ( R `  F ) )
 
Theoremtendo0cl 30109* The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
 
Theoremtendo0pl 30110* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  ( O P S )  =  S )
 
Theoremtendo0plr 30111* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  ( S P O )  =  S )
 
Theoremtendoicbv 30112* Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)
 |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   =>    |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g
 ) ) )
 
Theoremtendoi 30113* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( S  e.  E  ->  ( I `  S )  =  (
 g  e.  T  |->  `' ( S `  g
 ) ) )
 
Theoremtendoi2 30114* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( S  e.  E  /\  F  e.  T )  ->  (
 ( I `  S ) `  F )  =  `' ( S `  F ) )
 
Theoremtendoicl 30115* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E ) 
 ->  ( I `  S )  e.  E )
 
Theoremtendoipl 30116* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  B  =  ( Base `  K )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  (
 ( I `  S ) P S )  =  O )
 
Theoremtendoipl2 30117* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f
 ) ) )   &    |-  B  =  ( Base `  K )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  ->  ( S P ( I `  S ) )  =  O )
 
Theoremerngfset 30118* The division rings on trace-preserving endomorphisms for a lattice  K. (Contributed by NM, 8-Jun-2013.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 EDRing `  K )  =  ( w  e.  H  |->  {
 <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. , 
 <. ( +g  `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( (
 TEndo `  K ) `  w )  |->  ( f  e.  ( ( LTrn `  K ) `  w )  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w )  |->  ( s  o.  t ) ) >. } ) )
 
Theoremerngset 30119* The division ring on trace-preserving endomorphisms for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  D  =  { <. ( Base `  ndx ) ,  E >. , 
 <. ( +g  `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) >. } )
 
Theoremerngbase 30120 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom  W). TODO: the .t hypothesis isn't used. (Also look at others.) (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  C  =  ( Base `  D )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  C  =  E )
 
Theoremerngfplus 30121* Ring addition operation. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .+  =  ( +g  `  D )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) ) )
 
Theoremerngplus 30122* Ring addition operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .+  =  ( +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  ->  ( U 
 .+  V )  =  ( f  e.  T  |->  ( ( U `  f )  o.  ( V `  f ) ) ) )
 
Theoremerngplus2 30123 Ring addition operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .+  =  ( +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T ) )  ->  ( ( U  .+  V ) `  F )  =  (
 ( U `  F )  o.  ( V `  F ) ) )
 
Theoremerngfmul 30124* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .x.  =  ( .r `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) )
 
Theoremerngmul 30125 Ring addition operation. (Contributed by NM, 10-Jun-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .x.  =  ( .r `  D )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )
 )  ->  ( U  .x.  V )  =  ( U  o.  V ) )
 
Theoremerngfset-rN 30126* The division rings on trace-preserving endomorphisms for a lattice  K. (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 EDRing R `  K )  =  ( w  e.  H  |->  { <. ( Base `  ndx ) ,  ( ( TEndo `  K ) `  w ) >. ,  <. (
 +g  `  ndx ) ,  ( s  e.  (
 ( TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w )  |->  ( f  e.  ( (
 LTrn `  K ) `  w )  |->  ( ( s `  f )  o.  ( t `  f ) ) ) ) >. ,  <. ( .r
 `  ndx ) ,  (
 s  e.  ( (
 TEndo `  K ) `  w ) ,  t  e.  ( ( TEndo `  K ) `  w )  |->  ( t  o.  s ) ) >. } ) )
 
Theoremerngset-rN 30127* The division ring on trace-preserving endomorphisms for a fiducial co-atom  W. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  D  =  { <. ( Base `  ndx ) ,  E >. ,  <. ( +g  ` 
 ndx ) ,  (
 s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  ( t `  f ) ) ) ) >. ,  <. ( .r
 `  ndx ) ,  (
 s  e.  E ,  t  e.  E  |->  ( t  o.  s ) )
 >. } )
 
Theoremerngbase-rN 30128 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom  W). (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |-  C  =  ( Base `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  C  =  E )
 
Theoremerngfplus-rN 30129* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |- 
 .+  =  ( +g  `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .+  =  (
 s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  ( t `  f ) ) ) ) )
 
Theoremerngplus-rN 30130* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |- 
 .+  =  ( +g  `  D )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )
 )  ->  ( U  .+  V )  =  ( f  e.  T  |->  ( ( U `  f
 )  o.  ( V `
  f ) ) ) )
 
Theoremerngplus2-rN 30131 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |- 
 .+  =  ( +g  `  D )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T )
 )  ->  ( ( U  .+  V ) `  F )  =  (
 ( U `  F )  o.  ( V `  F ) ) )
 
Theoremerngfmul-rN 30132* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |- 
 .x.  =  ( .r `  D )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( t  o.  s ) ) )
 
Theoremerngmul-rN 30133 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |- 
 .x.  =  ( .r `  D )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )
 )  ->  ( U  .x.  V )  =  ( V  o.  U ) )
 
Theoremcdlemh1 30134 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  Q  e.  A )  /\  ( Q  .<_  ( P  .\/  ( R `  F ) )  /\  ( R `  F )  =/=  ( R `  G ) ) ) 
 ->  ( S  .\/  ( R `  ( G  o.  `' F ) ) )  =  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )
 
Theoremcdlemh2 30135 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( S  ./\  W )  =  .0.  )
 
Theoremcdlemh 30136 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Q  .\/  ( R `  ( G  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  Q  .<_  ( P 
 .\/  ( R `  F ) ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
 
Theoremcdlemi1 30137 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( U `  G ) `  P )  .<_  ( P  .\/  ( R `  G ) ) )
 
Theoremcdlemi2 30138 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( U `  G ) `  P )  .<_  ( ( ( U `  F ) `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )
 
Theoremcdlemi 30139 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  S  =  ( ( P  .\/  ( R `  G ) )  ./\  ( (
 ( U `  F ) `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( U  e.  E  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  G ) ) ) 
 ->  ( ( U `  G ) `  P )  =  S )
 
Theoremcdlemj1 30140 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T ) )  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `  F )  =/=  ( R `  g )  /\  ( R `
  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) ) 
 ->  ( ( U `  h ) `  p )  =  ( ( V `  h ) `  p ) )
 
Theoremcdlemj2 30141 Part of proof of Lemma J of [Crawley] p. 118. Eliminate  p. (Contributed by NM, 20-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T ) )  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `  F )  =/=  ( R `  g )  /\  ( R `
  g )  =/=  ( R `  h ) ) )  ->  ( U `  h )  =  ( V `  h ) )
 
Theoremcdlemj3 30142 Part of proof of Lemma J of [Crawley] p. 118. Eliminate  g. (Contributed by NM, 20-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T ) )  /\  h  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) )
 
Theoremtendocan 30143 Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) ) 
 ->  U  =  V )
 
Theoremtendoid0 30144* A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  (
 ( U `  F )  =  (  _I  |`  B )  <->  U  =  O ) )
 
Theoremtendo0mul 30145* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  ->  ( O  o.  U )  =  O )
 
Theoremtendo0mulr 30146* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  ->  ( U  o.  O )  =  O )
 
Theoremtendo1ne0 30147* The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  O )
 
Theoremtendoconid 30148* The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O ) )  ->  ( U  o.  V )  =/=  O )
 
Theoremtendotr 30149* The trace of the value of a non-zero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  =  ( R `  F ) )
 
Theoremcdlemk1 30150 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T ) 
 /\  ( ( R `
  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  .\/  ( N `
  P ) )  =  ( ( F `
  P )  .\/  ( R `  F ) ) )
 
Theoremcdlemk2 30151 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( G `  P )  .\/  ( R `  ( G  o.  `' F ) ) )  =  ( ( F `  P )  .\/  ( R `
  ( G  o.  `' F ) ) ) )
 
Theoremcdlemk3 30152 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( ( R `
  G )  =/=  ( R `  F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  (
 ( ( F `  P )  .\/  ( R `
  F ) ) 
 ./\  ( ( F `
  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  =  ( F `  P ) )
 
Theoremcdlemk4 30153 Part of proof of Lemma K of [Crawley] p. 118, last line. We use  X for their h, since  H is already used. (Contributed by NM, 24-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F ) ) ) )
 
Theoremcdlemk5a 30154 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T ) 
 /\  ( ( R `
  G )  =/=  ( R `  F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  (
 ( ( F `  P )  .\/  ( R `
  F ) ) 
 ./\  ( ( F `
  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F ) ) ) )
 
Theoremcdlemk5 30155 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( N  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( ( P  .\/  ( N `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F ) ) ) )
 
Theoremcdlemk6 30156 Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 29205. (Contributed by NM, 25-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( N  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  (
 ( R `  G )  =/=  ( R `  F )  /\  ( R `
  X )  =/=  ( R `  F ) ) ) ) 
 ->  ( ( P  .\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( ( ( G `
  P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
  P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )
 
Theoremcdlemk8 30157 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( G `  P )  .\/  ( X `  P ) )  =  ( ( G `  P )  .\/  ( R `
  ( X  o.  `' G ) ) ) )
 
Theoremcdlemk9 30158 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( ( G `  P )  .\/  ( X `
  P ) ) 
 ./\  W )  =  ( R `  ( X  o.  `' G ) ) )
 
Theoremcdlemk9bN 30159 Part of proof of Lemma K of [Crawley] p. 118. TODO: is this needed? If so, shorten with cdlemk9 30158 if that one is also needed. (Contributed by NM, 28-Jun-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( ( G `  P )  .\/  ( X `
  P ) ) 
 ./\  W )  =  ( R `  ( G  o.  `' X ) ) )
 
Theoremcdlemki 30160* Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 30164. (Contributed by NM, 25-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  I  =  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( G  o.  `' F ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  I  e.  T )
 
Theoremcdlemkvcl 30161 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  V  =  ( ( ( G `
  P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  P  e.  A ) 
 ->  V  e.  B )
 
Theoremcdlemk10 30162 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  V  =  ( ( ( G `
  P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  V  .<_  ( R `  ( X  o.  `' G ) ) )
 
Theoremcdlemksv 30163* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   =>    |-  ( G  e.  T  ->  ( S `  G )  =  ( iota_
 i  e.  T ( i `  P )  =  ( ( P 
 .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( G  o.  `' F ) ) ) ) ) )
 
Theoremcdlemksel 30164* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 30160? (Contributed by NM, 26-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) ) 
 ->  ( S `  G )  e.  T )
 
Theoremcdlemksat 30165* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) ) 
 ->  ( ( S `  G ) `  P )  e.  A )
 
Theoremcdlemksv2 30166* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function  S at the fixed  P parameter. (Contributed by NM, 26-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) ) 
 ->  ( ( S `  G ) `  P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( G  o.  `' F ) ) ) ) )
 
Theoremcdlemk7 30167* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  V  =  ( ( ( G `
  P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( ( N  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
  G )  =/=  ( R `  F )  /\  ( R `  X )  =/=  ( R `  F ) ) )  ->  ( ( S `  G ) `  P )  .<_  ( ( ( S `  X ) `  P )  .\/  V ) )
 
Theoremcdlemk11 30168* Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  V  =  ( ( ( G `
  P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( ( N  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
  G )  =/=  ( R `  F )  /\  ( R `  X )  =/=  ( R `  F ) ) )  ->  ( ( S `  G ) `  P )  .<_  ( ( ( S `  X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
 
Theoremcdlemk12 30169* Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  ./\  =  ( meet `  K )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  G  e.  T ) 
 /\  ( ( N  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( ( R `  G )  =/=  ( R `  F )  /\  ( R `
  X )  =/=  ( R `  F ) )  /\  ( R `
  G )  =/=  ( R `  X ) ) )  ->  ( ( S `  G ) `  P )  =  ( ( P  .\/  ( G `  P ) )  ./\  ( ( ( S `
  X ) `  P )  .\/  ( R `
  ( X  o.  `' G ) ) ) ) )
 
Theoremcdlemkoatnle 30170* Utility lemma. (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( O `  P )  e.  A  /\  -.  ( O `  P )  .<_  W ) )
 
Theoremcdlemk13 30171* Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119.  O,  D are k1, f1. (Contributed by NM, 1-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( O `  P )  =  ( ( P 
 .\/  ( R `  D ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( D  o.  `' F ) ) ) ) )
 
Theoremcdlemkole 30172* Utility lemma. (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( O `  P ) 
 .<_  ( P  .\/  ( R `  D ) ) )
 
Theoremcdlemk14 30173* Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119.  O,  D are k1, f1. (Contributed by NM, 1-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P ) 
 .<_  ( ( O `  P )  .\/  ( R `
  ( F  o.  `' D ) ) ) )
 
Theoremcdlemk15 30174* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119.  O,  D are k1, f1. (Contributed by NM, 1-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P ) 
 .<_  ( ( P  .\/  ( R `  F ) )  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D ) ) ) ) )
 
Theoremcdlemk16a 30175* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  G ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
 ( P  .\/  ( R `  G ) ) 
 ./\  ( ( O `
  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  e.  A  /\  -.  (
 ( P  .\/  ( R `  G ) ) 
 ./\  ( ( O `
  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  .<_  W ) )
 
Theoremcdlemk16 30176* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( ( P 
 .\/  ( R `  F ) )  ./\  ( ( O `  P )  .\/  ( R `
  ( F  o.  `' D ) ) ) )  e.  A  /\  -.  ( ( P  .\/  ( R `  F ) )  ./\  ( ( O `  P )  .\/  ( R `  ( F  o.  `' D ) ) ) )  .<_  W ) )
 
Theoremcdlemk17 30177* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119.  O,  D are k1, f1. (Contributed by NM, 1-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P )  =  ( ( P 
 .\/  ( R `  F ) )  ./\  ( ( O `  P )  .\/  ( R `
  ( F  o.  `' D ) ) ) ) )
 
Theoremcdlemk1u 30178* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( P  .\/  ( O `
  P ) ) 
 .<_  ( ( D `  P )  .\/  ( R `
  D ) ) )
 
Theoremcdlemk5auN 30179* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  ( ( R `  G )  =/=  ( R `  D )  /\  ( D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
 ( D `  P )  .\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `
  ( G  o.  `' D ) ) ) )  .<_  ( ( X `
  P )  .\/  ( R `  ( X  o.  `' D ) ) ) )
 
Theoremcdlemk5u 30180* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( ( N  e.  T  /\  G  e.  T  /\  X  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B ) ) 
 /\  ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X )  =/=  ( R `  D ) ) ) )  ->  ( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `
  ( G  o.  `' D ) ) ) )  .<_  ( ( X `
  P )  .\/  ( R `  ( X  o.  `' D ) ) ) )
 
Theoremcdlemk6u 30181* Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 29205. (Contributed by NM, 4-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( ( N  e.  T  /\  G  e.  T  /\  X  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B ) ) 
 /\  ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X )  =/=  ( R `  D ) ) ) )  ->  ( ( P  .\/  ( G `  P ) )  ./\  ( ( O `  P )  .\/  ( R `
  ( G  o.  `' D ) ) ) )  .<_  ( ( ( ( G `  P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' D ) )  .\/  ( R `
  ( X  o.  `' D ) ) ) )  .\/  ( (
 ( X `  P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' D ) )  .\/  ( O `
  P ) ) ) ) )
 
Theoremcdlemkj 30182* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  Z  =  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  G ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  e.  T )
 
TheoremcdlemkuvN 30183* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma1 (p) function  U. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( G  e.  T  ->  ( U `  G )  =  ( iota_ j  e.  T ( j `  P )  =  (
 ( P  .\/  ( R `  G ) ) 
 ./\  ( ( O `
  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) ) )
 
Theoremcdlemkuel 30184* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma1 (p) function to be a translation. TODO: combine cdlemkj 30182? (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  G ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  G )  e.  T )
 
Theoremcdlemkuat 30185* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  G ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  e.  A )
 
Theoremcdlemkuv2 30186* Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma1 (p) is  U, f1 is  D, and k1 is  O. (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
  D )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  G ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  =  (
 ( P  .\/  ( R `  G ) ) 
 ./\  ( ( O `
  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
 
Theoremcdlemk18 30187* Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119.  N,  U,  O,  D are k, sigma1 (p), k1, f1. (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( N `  P )  =  ( ( U `
  F ) `  P ) )
 
Theoremcdlemk19 30188* Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119.  N,  U,  O,  D are k, sigma1 (p), k1, f1. (Contributed by NM, 2-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R `  F )  =  ( R `  N ) ) 
 /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( U `  F )  =  N )
 
Theoremcdlemk7u 30189* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma1 case. (Contributed by NM, 3-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   &    |-  V  =  ( ( ( G `  P )  .\/  ( X `
  P ) ) 
 ./\  ( ( R `
  ( G  o.  `' D ) )  .\/  ( R `  ( X  o.  `' D ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  D  e.  T ) 
 /\  ( ( N  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `
  X )  =/=  ( R `  D ) ) ) ) 
 ->  ( ( U `  G ) `  P )  .<_  ( ( ( U `  X ) `
  P )  .\/  V ) )
 
Theoremcdlemk11u 30190* Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma1 ( U) case. (Contributed by NM, 4-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   &    |-  V  =  ( ( ( G `  P )  .\/  ( X `
  P ) ) 
 ./\  ( ( R `
  ( G  o.  `' D ) )  .\/  ( R `  ( X  o.  `' D ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  F  e.  T  /\  D  e.  T ) 
 /\  ( ( N  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `
  X )  =/=  ( R `  D ) ) ) ) 
 ->  ( ( U `  G ) `  P )  .<_  ( ( ( U `  X ) `
  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
 
Theoremcdlemk12u 30191* Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma1 ( U) case. (Contributed by NM, 4-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `
  ( f  o.  `' F ) ) ) ) ) )   &    |-  O  =  ( S `  D )   &    |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
  P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( ( O `  P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( ( N  e.  T  /\  G  e.  T  /\  X  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `
  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) 
 /\  G  =/=  (  _I  |`  B ) ) 
 /\  ( X  =/=  (  _I  |`  B )  /\