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Theorem dihatexv 34981
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 2442 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
7 dihatexv.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 dihatexv.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 eqid 2442 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2442 . . . . . . . . 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 34884 . . . . . . . 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 1216 . . . . . . 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 2442 . . . . . . . . . . . . . 14  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
195, 8, 9, 18, 10tendo0cl 34432 . . . . . . . . . . . . 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 34731 . . . . . . . . . . . 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 1216 . . . . . . . . . . 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 3886 . . . . . . . . . . . . . 14  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  { x }  =  { <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >. } )
2524fveq2d 5694 . . . . . . . . . . . . 13  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  ( N `  {
x } )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) >. } ) )
2625eqeq2d 2453 . . . . . . . . . . . 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 3072 . . . . . . . . . . 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 2840 . . . . . . 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 2442 . . . . . . . . . . 11  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
356, 7, 8, 34lhpocnel2 33661 . . . . . . . . . 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 2442 . . . . . . . . . 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 34221 . . . . . . . . 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 1222 . . . . . . . 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 34411 . . . . . . . . 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 34731 . . . . . . . 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 1216 . . . . . . 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 34885 . . . . . . . 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 1216 . . . . . . 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 3886 . . . . . . . . . 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 5694 . . . . . . . . 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 2453 . . . . . . . 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 3072 . . . . . . 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 33003 . . . . . . . . . . 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 2442 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
6059, 7atn0 32951 . . . . . . . . . 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 3886 . . . . . . . . . . . . . . . 16  |-  ( x  =  .0.  ->  { x }  =  {  .0.  } )
6362fveq2d 5694 . . . . . . . . . . . . . . 15  |-  ( x  =  .0.  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
64633ad2ant3 1011 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
65 simp1ll 1051 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ph )
668, 11, 1dvhlmod 34753 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  LMod )
67 dihatexv.o . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  U )
6867, 13lspsn0 17088 . . . . . . . . . . . . . . 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 2474 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  {  .0.  } )
71 simp2 989 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  Q )  =  ( N `  { x } ) )
7259, 8, 12, 11, 67dih0 34923 . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . 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 33005 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  OP )
805, 59op0cl 32827 . . . . . . . . . . . . . 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 34908 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 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 1189 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  -> 
( x  =  .0. 
->  Q  =  ( 0. `  K ) ) )
8685necon3d 2645 . . . . . . . . 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 2827 . . . . 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 34915 . . . . . . . 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 5690 . . . . . . . 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 2476 . . . . . 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 2442 . . . . . . . . 9  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
10421, 13, 67, 103lsatlspsn2 32635 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  x  e.  V  /\  x  =/=  .0.  )  ->  ( N `  { x } )  e.  (LSAtoms `  U ) )
105100, 101, 102, 104syl3anc 1218 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( N `  {
x } )  e.  (LSAtoms `  U )
)
1067, 8, 11, 12, 103dihlatat 34980 . . . . . . 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 2516 . . . . 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 2840 . . 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 4003 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715    \ cdif 3324   {csn 3876   <.cop 3882   class class class wbr 4291    e. cmpt 4349    _I cid 4630   `'ccnv 4838    |` cres 4841   ` cfv 5417   iota_crio 6050   Basecbs 14173   lecple 14244   occoc 14245   0gc0g 14377   0.cp0 15206   LModclmod 16947   LSpanclspn 17051  LSAtomsclsa 32617   OPcops 32815   Atomscatm 32906   AtLatcal 32907   HLchlt 32993   LHypclh 33626   LTrncltrn 33743   TEndoctendo 34394   DVecHcdvh 34721   DIsoHcdih 34871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-riotaBAD 32602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6791  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-0g 14379  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-subg 15677  df-cntz 15834  df-lsm 16134  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-drng 16833  df-lmod 16949  df-lss 17013  df-lsp 17052  df-lvec 17183  df-lsatoms 32619  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-lplanes 33141  df-lvols 33142  df-lines 33143  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801  df-tendo 34397  df-edring 34399  df-disoa 34672  df-dvech 34722  df-dib 34782  df-dic 34816  df-dih 34872
This theorem is referenced by:  dihatexv2  34982
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