Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihatexv Structured version   Unicode version

Theorem dihatexv 36012
Description: There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
Hypotheses
Ref Expression
dihatexv.b  |-  B  =  ( Base `  K
)
dihatexv.a  |-  A  =  ( Atoms `  K )
dihatexv.h  |-  H  =  ( LHyp `  K
)
dihatexv.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihatexv.v  |-  V  =  ( Base `  U
)
dihatexv.o  |-  .0.  =  ( 0g `  U )
dihatexv.n  |-  N  =  ( LSpan `  U )
dihatexv.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihatexv.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dihatexv.q  |-  ( ph  ->  Q  e.  B )
Assertion
Ref Expression
dihatexv  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V 
\  {  .0.  }
) ( I `  Q )  =  ( N `  { x } ) ) )
Distinct variable groups:    x, A    x, B    x, I    x, K    x, N    x, Q    x, V    x, W    ph, x
Allowed substitution hints:    U( x)    H( x)    .0. ( x)

Proof of Theorem dihatexv
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihatexv.k . . . . . . . . 9  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
21ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  -> 
( K  e.  HL  /\  W  e.  H ) )
3 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  Q  e.  A )
4 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  Q ( le `  K ) W )
5 dihatexv.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
6 eqid 2462 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
7 dihatexv.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 dihatexv.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 eqid 2462 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2462 . . . . . . . . 9  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
11 dihatexv.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
12 dihatexv.i . . . . . . . . 9  |-  I  =  ( ( DIsoH `  K
) `  W )
13 dihatexv.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
145, 6, 7, 8, 9, 10, 11, 12, 13dih1dimb2 35915 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q ( le `  K ) W ) )  ->  E. g  e.  (
( LTrn `  K ) `  W ) ( g  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >. } ) ) )
152, 3, 4, 14syl12anc 1221 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  E. g  e.  (
( LTrn `  K ) `  W ) ( g  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >. } ) ) )
161ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  g  e.  ( ( LTrn `  K
) `  W )
)
18 eqid 2462 . . . . . . . . . . . . . 14  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
195, 8, 9, 18, 10tendo0cl 35463 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) )
2016, 19syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) )
21 dihatexv.v . . . . . . . . . . . . 13  |-  V  =  ( Base `  U
)
228, 9, 18, 11, 21dvhelvbasei 35762 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( g  e.  ( ( LTrn `  K
) `  W )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) ) )  ->  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  e.  V )
2316, 17, 20, 22syl12anc 1221 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >.  e.  V
)
24 sneq 4032 . . . . . . . . . . . . . 14  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  { x }  =  { <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >. } )
2524fveq2d 5863 . . . . . . . . . . . . 13  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  ( N `  {
x } )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) >. } ) )
2625eqeq2d 2476 . . . . . . . . . . . 12  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  ( ( I `  Q )  =  ( N `  { x } )  <->  ( I `  Q )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >. } ) ) )
2726rspcev 3209 . . . . . . . . . . 11  |-  ( (
<. g ,  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) >.  e.  V  /\  ( I `  Q
)  =  ( N `
 { <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >. } ) )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
2823, 27sylan 471 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le
`  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  /\  ( I `  Q )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >. } ) )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
2928ex 434 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
I `  Q )  =  ( N `  { <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >. } )  ->  E. x  e.  V  ( I `  Q )  =  ( N `  { x } ) ) )
3029adantld 467 . . . . . . . 8  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
g  =/=  (  _I  |`  B )  /\  (
I `  Q )  =  ( N `  { <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >. } ) )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) ) )
3130rexlimdva 2950 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  -> 
( E. g  e.  ( ( LTrn `  K
) `  W )
( g  =/=  (  _I  |`  B )  /\  ( I `  Q
)  =  ( N `
 { <. g ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) >. } ) )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) ) )
3215, 31mpd 15 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
331ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( K  e.  HL  /\  W  e.  H ) )
34 eqid 2462 . . . . . . . . . . 11  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
356, 7, 8, 34lhpocnel2 34692 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
3633, 35syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )
( le `  K
) W ) )
37 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  Q  e.  A
)
38 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  -.  Q ( le `  K ) W )
39 eqid 2462 . . . . . . . . . 10  |-  ( iota_ f  e.  ( ( LTrn `  K ) `  W
) ( f `  ( ( oc `  K ) `  W
) )  =  Q )  =  ( iota_ f  e.  ( ( LTrn `  K ) `  W
) ( f `  ( ( oc `  K ) `  W
) )  =  Q )
406, 7, 8, 9, 39ltrniotacl 35252 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )
( le `  K
) W )  /\  ( Q  e.  A  /\  -.  Q ( le
`  K ) W ) )  ->  ( iota_ f  e.  ( (
LTrn `  K ) `  W ) ( f `
 ( ( oc
`  K ) `  W ) )  =  Q )  e.  ( ( LTrn `  K
) `  W )
)
4133, 36, 37, 38, 40syl112anc 1227 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q )  e.  ( ( LTrn `  K ) `  W
) )
428, 9, 18tendoidcl 35442 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  (
( LTrn `  K ) `  W ) )  e.  ( ( TEndo `  K
) `  W )
)
4333, 42syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  (  _I  |`  (
( LTrn `  K ) `  W ) )  e.  ( ( TEndo `  K
) `  W )
)
448, 9, 18, 11, 21dvhelvbasei 35762 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( iota_ f  e.  ( ( LTrn `  K ) `  W
) ( f `  ( ( oc `  K ) `  W
) )  =  Q )  e.  ( (
LTrn `  K ) `  W )  /\  (  _I  |`  ( ( LTrn `  K ) `  W
) )  e.  ( ( TEndo `  K ) `  W ) ) )  ->  <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.  e.  V )
4533, 41, 43, 44syl12anc 1221 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.  e.  V )
466, 7, 8, 34, 9, 12, 11, 13, 39dih1dimc 35916 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q
( le `  K
) W ) )  ->  ( I `  Q )  =  ( N `  { <. (
iota_ f  e.  (
( LTrn `  K ) `  W ) ( f `
 ( ( oc
`  K ) `  W ) )  =  Q ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } ) )
4733, 37, 38, 46syl12anc 1221 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( I `  Q )  =  ( N `  { <. (
iota_ f  e.  (
( LTrn `  K ) `  W ) ( f `
 ( ( oc
`  K ) `  W ) )  =  Q ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } ) )
48 sneq 4032 . . . . . . . . . 10  |-  ( x  =  <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.  ->  { x }  =  { <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >. } )
4948fveq2d 5863 . . . . . . . . 9  |-  ( x  =  <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.  ->  ( N `  {
x } )  =  ( N `  { <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >. } ) )
5049eqeq2d 2476 . . . . . . . 8  |-  ( x  =  <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.  ->  ( ( I `  Q )  =  ( N `  { x } )  <->  ( I `  Q )  =  ( N `  { <. (
iota_ f  e.  (
( LTrn `  K ) `  W ) ( f `
 ( ( oc
`  K ) `  W ) )  =  Q ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } ) ) )
5150rspcev 3209 . . . . . . 7  |-  ( (
<. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.  e.  V  /\  (
I `  Q )  =  ( N `  { <. ( iota_ f  e.  ( ( LTrn `  K
) `  W )
( f `  (
( oc `  K
) `  W )
)  =  Q ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >. } ) )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
5245, 47, 51syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
5332, 52pm2.61dan 789 . . . . 5  |-  ( (
ph  /\  Q  e.  A )  ->  E. x  e.  V  ( I `  Q )  =  ( N `  { x } ) )
541simpld 459 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
5554ad3antrrr 729 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  K  e.  HL )
56 hlatl 34034 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  AtLat )
5755, 56syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  K  e.  AtLat )
58 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  e.  A )
59 eqid 2462 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
6059, 7atn0 33982 . . . . . . . . . 10  |-  ( ( K  e.  AtLat  /\  Q  e.  A )  ->  Q  =/=  ( 0. `  K
) )
6157, 58, 60syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  =/=  ( 0. `  K ) )
62 sneq 4032 . . . . . . . . . . . . . . . 16  |-  ( x  =  .0.  ->  { x }  =  {  .0.  } )
6362fveq2d 5863 . . . . . . . . . . . . . . 15  |-  ( x  =  .0.  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
64633ad2ant3 1014 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
65 simp1ll 1054 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ph )
668, 11, 1dvhlmod 35784 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  LMod )
67 dihatexv.o . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  U )
6867, 13lspsn0 17432 . . . . . . . . . . . . . . 15  |-  ( U  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
6965, 66, 683syl 20 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  {  .0.  }
)  =  {  .0.  } )
7064, 69eqtrd 2503 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  {  .0.  } )
71 simp2 992 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  Q )  =  ( N `  { x } ) )
7259, 8, 12, 11, 67dih0 35954 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  ( 0. `  K ) )  =  {  .0.  }
)
7365, 1, 723syl 20 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  ( 0. `  K ) )  =  {  .0.  } )
7470, 71, 733eqtr4d 2513 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  Q )  =  ( I `  ( 0. `  K ) ) )
7565, 1syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
76 dihatexv.q . . . . . . . . . . . . . 14  |-  ( ph  ->  Q  e.  B )
7765, 76syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  Q  e.  B )
7865, 54syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  K  e.  HL )
79 hlop 34036 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  OP )
805, 59op0cl 33858 . . . . . . . . . . . . . 14  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
8178, 79, 803syl 20 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( 0. `  K )  e.  B )
825, 8, 12dih11 35939 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B  /\  ( 0. `  K
)  e.  B )  ->  ( ( I `
 Q )  =  ( I `  ( 0. `  K ) )  <-> 
Q  =  ( 0.
`  K ) ) )
8375, 77, 81, 82syl3anc 1223 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
( I `  Q
)  =  ( I `
 ( 0. `  K ) )  <->  Q  =  ( 0. `  K ) ) )
8474, 83mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  Q  =  ( 0. `  K ) )
85843expia 1193 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  -> 
( x  =  .0. 
->  Q  =  ( 0. `  K ) ) )
8685necon3d 2686 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  -> 
( Q  =/=  ( 0. `  K )  ->  x  =/=  .0.  ) )
8761, 86mpd 15 . . . . . . . 8  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  x  =/=  .0.  )
8887ex 434 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V )  ->  (
( I `  Q
)  =  ( N `
 { x }
)  ->  x  =/=  .0.  ) )
8988ancrd 554 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V )  ->  (
( I `  Q
)  =  ( N `
 { x }
)  ->  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) ) )
9089reximdva 2933 . . . . 5  |-  ( (
ph  /\  Q  e.  A )  ->  ( E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
)  ->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) ) )
9153, 90mpd 15 . . . 4  |-  ( (
ph  /\  Q  e.  A )  ->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) )
9291ex 434 . . 3  |-  ( ph  ->  ( Q  e.  A  ->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) ) )
931ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
9476ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  e.  B )
955, 8, 12dihcnvid1 35946 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( `' I `  ( I `  Q ) )  =  Q )
9693, 94, 95syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( I `  Q
) )  =  Q )
97 fveq2 5859 . . . . . . . 8  |-  ( ( I `  Q )  =  ( N `  { x } )  ->  ( `' I `  ( I `  Q
) )  =  ( `' I `  ( N `
 { x }
) ) )
9897ad2antll 728 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( I `  Q
) )  =  ( `' I `  ( N `
 { x }
) ) )
9996, 98eqtr3d 2505 . . . . . 6  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  =  ( `' I `  ( N `  { x } ) ) )
10066ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  U  e.  LMod )
101 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  x  e.  V )
102 simprl 755 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  x  =/=  .0.  )
103 eqid 2462 . . . . . . . . 9  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
10421, 13, 67, 103lsatlspsn2 33666 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  x  e.  V  /\  x  =/=  .0.  )  ->  ( N `  { x } )  e.  (LSAtoms `  U ) )
105100, 101, 102, 104syl3anc 1223 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( N `  {
x } )  e.  (LSAtoms `  U )
)
1067, 8, 11, 12, 103dihlatat 36011 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  (LSAtoms `  U
) )  ->  ( `' I `  ( N `
 { x }
) )  e.  A
)
10793, 105, 106syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( N `  { x } ) )  e.  A )
10899, 107eqeltrd 2550 . . . . 5  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  e.  A )
109108ex 434 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) )  ->  Q  e.  A ) )
110109rexlimdva 2950 . . 3  |-  ( ph  ->  ( E. x  e.  V  ( x  =/= 
.0.  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  e.  A )
)
11192, 110impbid 191 . 2  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) ) )
112 rexdifsn 4151 . 2  |-  ( E. x  e.  ( V 
\  {  .0.  }
) ( I `  Q )  =  ( N `  { x } )  <->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) )
113111, 112syl6bbr 263 1  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V 
\  {  .0.  }
) ( I `  Q )  =  ( N `  { x } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810    \ cdif 3468   {csn 4022   <.cop 4028   class class class wbr 4442    |-> cmpt 4500    _I cid 4785   `'ccnv 4993    |` cres 4996   ` cfv 5581   iota_crio 6237   Basecbs 14481   lecple 14553   occoc 14554   0gc0g 14686   0.cp0 15515   LModclmod 17290   LSpanclspn 17395  LSAtomsclsa 33648   OPcops 33846   Atomscatm 33937   AtLatcal 33938   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   TEndoctendo 35425   DVecHcdvh 35752   DIsoHcdih 35902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-riotaBAD 33633
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-undef 6994  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-0g 14688  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-sbg 15855  df-subg 15988  df-cntz 16145  df-lsm 16447  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-dvr 17111  df-drng 17176  df-lmod 17292  df-lss 17357  df-lsp 17396  df-lvec 17527  df-lsatoms 33650  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832  df-tendo 35428  df-edring 35430  df-disoa 35703  df-dvech 35753  df-dib 35813  df-dic 35847  df-dih 35903
This theorem is referenced by:  dihatexv2  36013
  Copyright terms: Public domain W3C validator