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Theorem dihatexv 31453
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 707 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  -> 
( K  e.  HL  /\  W  e.  H ) )
3 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  Q  e.  A )
4 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  Q ( le `  K ) W )
5 dihatexv.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
6 eqid 2387 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
7 dihatexv.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 dihatexv.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 eqid 2387 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2387 . . . . . . . . 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 31356 . . . . . . . 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 1182 . . . . . . 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 711 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  g  e.  ( ( LTrn `  K
) `  W )
)
18 eqid 2387 . . . . . . . . . . . . . 14  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
195, 8, 9, 18, 10tendo0cl 30904 . . . . . . . . . . . . 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 31203 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 3768 . . . . . . . . . . . . . 14  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  { x }  =  { <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >. } )
2524fveq2d 5672 . . . . . . . . . . . . 13  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  ( N `  {
x } )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) >. } ) )
2625eqeq2d 2398 . . . . . . . . . . . 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 2995 . . . . . . . . . . 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 458 . . . . . . . . . 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 424 . . . . . . . . 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 454 . . . . . . . 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 2773 . . . . . . 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 707 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( K  e.  HL  /\  W  e.  H ) )
34 eqid 2387 . . . . . . . . . . 11  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
356, 7, 8, 34lhpocnel2 30133 . . . . . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  Q  e.  A
)
38 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  -.  Q ( le `  K ) W )
39 eqid 2387 . . . . . . . . . 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 30693 . . . . . . . . 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 1188 . . . . . . . 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 30883 . . . . . . . . 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 31203 . . . . . . . 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 1182 . . . . . . 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 31357 . . . . . . . 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 1182 . . . . . . 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 3768 . . . . . . . . . 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 5672 . . . . . . . . 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 2398 . . . . . . . 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 2995 . . . . . . 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 643 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
5332, 52pm2.61dan 767 . . . . 5  |-  ( (
ph  /\  Q  e.  A )  ->  E. x  e.  V  ( I `  Q )  =  ( N `  { x } ) )
541simpld 446 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
5554ad3antrrr 711 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  K  e.  HL )
56 hlatl 29475 . . . . . . . . . . 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 736 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  e.  A )
59 eqid 2387 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
6059, 7atn0 29423 . . . . . . . . . 10  |-  ( ( K  e.  AtLat  /\  Q  e.  A )  ->  Q  =/=  ( 0. `  K
) )
6157, 58, 60syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  =/=  ( 0. `  K ) )
62 sneq 3768 . . . . . . . . . . . . . . . 16  |-  ( x  =  .0.  ->  { x }  =  {  .0.  } )
6362fveq2d 5672 . . . . . . . . . . . . . . 15  |-  ( x  =  .0.  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
64633ad2ant3 980 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
65 simp1ll 1020 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ph )
668, 11, 1dvhlmod 31225 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  LMod )
67 dihatexv.o . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  U )
6867, 13lspsn0 16011 . . . . . . . . . . . . . . 15  |-  ( U  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
6965, 66, 683syl 19 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  {  .0.  }
)  =  {  .0.  } )
7064, 69eqtrd 2419 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  {  .0.  } )
71 simp2 958 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  Q )  =  ( N `  { x } ) )
7259, 8, 12, 11, 67dih0 31395 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  ( 0. `  K ) )  =  {  .0.  }
)
7365, 1, 723syl 19 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  ( 0. `  K ) )  =  {  .0.  } )
7470, 71, 733eqtr4d 2429 . . . . . . . . . . . 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 29477 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  OP )
805, 59op0cl 29299 . . . . . . . . . . . . . 14  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
8178, 79, 803syl 19 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( 0. `  K )  e.  B )
825, 8, 12dih11 31380 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
( I `  Q
)  =  ( I `
 ( 0. `  K ) )  <->  Q  =  ( 0. `  K ) ) )
8474, 83mpbid 202 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  Q  =  ( 0. `  K ) )
85843expia 1155 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  -> 
( x  =  .0. 
->  Q  =  ( 0. `  K ) ) )
8685necon3d 2588 . . . . . . . . 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 424 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V )  ->  (
( I `  Q
)  =  ( N `
 { x }
)  ->  x  =/=  .0.  ) )
8988ancrd 538 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V )  ->  (
( I `  Q
)  =  ( N `
 { x }
)  ->  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) ) )
9089reximdva 2761 . . . . 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 424 . . 3  |-  ( ph  ->  ( Q  e.  A  ->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) ) )
931ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
9476ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  e.  B )
955, 8, 12dihcnvid1 31387 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( `' I `  ( I `  Q ) )  =  Q )
9693, 94, 95syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( I `  Q
) )  =  Q )
97 fveq2 5668 . . . . . . . 8  |-  ( ( I `  Q )  =  ( N `  { x } )  ->  ( `' I `  ( I `  Q
) )  =  ( `' I `  ( N `
 { x }
) ) )
9897ad2antll 710 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( I `  Q
) )  =  ( `' I `  ( N `
 { x }
) ) )
9996, 98eqtr3d 2421 . . . . . 6  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  =  ( `' I `  ( N `  { x } ) ) )
10066ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  U  e.  LMod )
101 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  x  e.  V )
102 simprl 733 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  x  =/=  .0.  )
103 eqid 2387 . . . . . . . . 9  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
10421, 13, 67, 103lsatlspsn2 29107 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  x  e.  V  /\  x  =/=  .0.  )  ->  ( N `  { x } )  e.  (LSAtoms `  U ) )
105100, 101, 102, 104syl3anc 1184 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( N `  {
x } )  e.  (LSAtoms `  U )
)
1067, 8, 11, 12, 103dihlatat 31452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  (LSAtoms `  U
) )  ->  ( `' I `  ( N `
 { x }
) )  e.  A
)
10793, 105, 106syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( N `  { x } ) )  e.  A )
10899, 107eqeltrd 2461 . . . . 5  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  e.  A )
109108ex 424 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) )  ->  Q  e.  A ) )
110109rexlimdva 2773 . . 3  |-  ( ph  ->  ( E. x  e.  V  ( x  =/= 
.0.  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  e.  A )
)
11192, 110impbid 184 . 2  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) ) )
112 rexdifsn 3874 . 2  |-  ( E. x  e.  ( V 
\  {  .0.  }
) ( I `  Q )  =  ( N `  { x } )  <->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) )
113111, 112syl6bbr 255 1  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V 
\  {  .0.  }
) ( I `  Q )  =  ( N `  { x } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    \ cdif 3260   {csn 3757   <.cop 3760   class class class wbr 4153    e. cmpt 4207    _I cid 4434   `'ccnv 4817    |` cres 4820   ` cfv 5394   iota_crio 6478   Basecbs 13396   lecple 13463   occoc 13464   0gc0g 13650   0.cp0 14393   LModclmod 15877   LSpanclspn 15974  LSAtomsclsa 29089   OPcops 29287   Atomscatm 29378   AtLatcal 29379   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   TEndoctendo 30866   DVecHcdvh 31193   DIsoHcdih 31343
This theorem is referenced by:  dihatexv2  31454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-undef 6479  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-0g 13654  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-cntz 15043  df-lsm 15197  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-dvr 15715  df-drng 15764  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lvec 16102  df-lsatoms 29091  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tendo 30869  df-edring 30871  df-disoa 31144  df-dvech 31194  df-dib 31254  df-dic 31288  df-dih 31344
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