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Theorem dihatexv 32150
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 706 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  -> 
( K  e.  HL  /\  W  e.  H ) )
3 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  Q  e.  A )
4 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  Q ( le `  K ) W )
5 dihatexv.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
6 eqid 2296 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
7 dihatexv.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 dihatexv.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 eqid 2296 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2296 . . . . . . . . 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 32053 . . . . . . . 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 1180 . . . . . . 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 710 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  Q ( le `  K ) W )  /\  g  e.  ( ( LTrn `  K
) `  W )
)  ->  g  e.  ( ( LTrn `  K
) `  W )
)
18 eqid 2296 . . . . . . . . . . . . . 14  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
195, 8, 9, 18, 10tendo0cl 31601 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) )
2016, 19syl 15 . . . . . . . . . . . 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 31900 . . . . . . . . . . . 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 1180 . . . . . . . . . . 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 3664 . . . . . . . . . . . . . 14  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  { x }  =  { <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >. } )
2524fveq2d 5545 . . . . . . . . . . . . 13  |-  ( x  =  <. g ,  ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) >.  ->  ( N `  {
x } )  =  ( N `  { <. g ,  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) >. } ) )
2625eqeq2d 2307 . . . . . . . . . . . 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 2897 . . . . . . . . . . 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 457 . . . . . . . . . 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 423 . . . . . . . . 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 453 . . . . . . . 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 2680 . . . . . . 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 14 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  Q
( le `  K
) W )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
331ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( K  e.  HL  /\  W  e.  H ) )
34 eqid 2296 . . . . . . . . . . 11  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
356, 7, 8, 34lhpocnel2 30830 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
3633, 35syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )
( le `  K
) W ) )
37 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  Q  e.  A
)
38 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  -.  Q ( le `  K ) W )
39 eqid 2296 . . . . . . . . . 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 31390 . . . . . . . . 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 1186 . . . . . . . 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 31580 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  (
( LTrn `  K ) `  W ) )  e.  ( ( TEndo `  K
) `  W )
)
4333, 42syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  (  _I  |`  (
( LTrn `  K ) `  W ) )  e.  ( ( TEndo `  K
) `  W )
)
448, 9, 18, 11, 21dvhelvbasei 31900 . . . . . . . 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 1180 . . . . . . 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 32054 . . . . . . . 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 1180 . . . . . . 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 3664 . . . . . . . . . 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 5545 . . . . . . . . 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 2307 . . . . . . . 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 2897 . . . . . . 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 642 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  -.  Q ( le `  K ) W )  ->  E. x  e.  V  ( I `  Q
)  =  ( N `
 { x }
) )
5332, 52pm2.61dan 766 . . . . 5  |-  ( (
ph  /\  Q  e.  A )  ->  E. x  e.  V  ( I `  Q )  =  ( N `  { x } ) )
541simpld 445 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
5554ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  K  e.  HL )
56 hlatl 30172 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  AtLat )
5755, 56syl 15 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  K  e.  AtLat )
58 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  e.  A )
59 eqid 2296 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
6059, 7atn0 30120 . . . . . . . . . 10  |-  ( ( K  e.  AtLat  /\  Q  e.  A )  ->  Q  =/=  ( 0. `  K
) )
6157, 58, 60syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  =/=  ( 0. `  K ) )
62 sneq 3664 . . . . . . . . . . . . . . . 16  |-  ( x  =  .0.  ->  { x }  =  {  .0.  } )
6362fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( x  =  .0.  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
64633ad2ant3 978 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  ( N `  {  .0.  } ) )
65 simp1ll 1018 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ph )
668, 11, 1dvhlmod 31922 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  LMod )
67 dihatexv.o . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  U )
6867, 13lspsn0 15781 . . . . . . . . . . . . . . 15  |-  ( U  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
6965, 66, 683syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  {  .0.  }
)  =  {  .0.  } )
7064, 69eqtrd 2328 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( N `  { x } )  =  {  .0.  } )
71 simp2 956 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  Q )  =  ( N `  { x } ) )
7259, 8, 12, 11, 67dih0 32092 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  ( 0. `  K ) )  =  {  .0.  }
)
7365, 1, 723syl 18 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  ( 0. `  K ) )  =  {  .0.  } )
7470, 71, 733eqtr4d 2338 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
I `  Q )  =  ( I `  ( 0. `  K ) ) )
7565, 1syl 15 . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  Q  e.  B )
7865, 54syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  K  e.  HL )
79 hlop 30174 . . . . . . . . . . . . . 14  |-  ( K  e.  HL  ->  K  e.  OP )
805, 59op0cl 29996 . . . . . . . . . . . . . 14  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  B )
8178, 79, 803syl 18 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  ( 0. `  K )  e.  B )
825, 8, 12dih11 32077 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  (
( I `  Q
)  =  ( I `
 ( 0. `  K ) )  <->  Q  =  ( 0. `  K ) ) )
8474, 83mpbid 201 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } )  /\  x  =  .0.  )  ->  Q  =  ( 0. `  K ) )
85843expia 1153 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  -> 
( x  =  .0. 
->  Q  =  ( 0. `  K ) ) )
8685necon3d 2497 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  -> 
( Q  =/=  ( 0. `  K )  ->  x  =/=  .0.  ) )
8761, 86mpd 14 . . . . . . . 8  |-  ( ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V
)  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  x  =/=  .0.  )
8887ex 423 . . . . . . 7  |-  ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V )  ->  (
( I `  Q
)  =  ( N `
 { x }
)  ->  x  =/=  .0.  ) )
8988ancrd 537 . . . . . 6  |-  ( ( ( ph  /\  Q  e.  A )  /\  x  e.  V )  ->  (
( I `  Q
)  =  ( N `
 { x }
)  ->  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) ) )
9089reximdva 2668 . . . . 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 14 . . . 4  |-  ( (
ph  /\  Q  e.  A )  ->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) )
9291ex 423 . . 3  |-  ( ph  ->  ( Q  e.  A  ->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) ) )
931ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
9476ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  e.  B )
955, 8, 12dihcnvid1 32084 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( `' I `  ( I `  Q ) )  =  Q )
9693, 94, 95syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( I `  Q
) )  =  Q )
97 fveq2 5541 . . . . . . . 8  |-  ( ( I `  Q )  =  ( N `  { x } )  ->  ( `' I `  ( I `  Q
) )  =  ( `' I `  ( N `
 { x }
) ) )
9897ad2antll 709 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( I `  Q
) )  =  ( `' I `  ( N `
 { x }
) ) )
9996, 98eqtr3d 2330 . . . . . 6  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  =  ( `' I `  ( N `  { x } ) ) )
10066ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  U  e.  LMod )
101 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  x  e.  V )
102 simprl 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  x  =/=  .0.  )
103 eqid 2296 . . . . . . . . 9  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
10421, 13, 67, 103lsatlspsn2 29804 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  x  e.  V  /\  x  =/=  .0.  )  ->  ( N `  { x } )  e.  (LSAtoms `  U ) )
105100, 101, 102, 104syl3anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( N `  {
x } )  e.  (LSAtoms `  U )
)
1067, 8, 11, 12, 103dihlatat 32149 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { x } )  e.  (LSAtoms `  U
) )  ->  ( `' I `  ( N `
 { x }
) )  e.  A
)
10793, 105, 106syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  -> 
( `' I `  ( N `  { x } ) )  e.  A )
10899, 107eqeltrd 2370 . . . . 5  |-  ( ( ( ph  /\  x  e.  V )  /\  (
x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) )  ->  Q  e.  A )
109108ex 423 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) )  ->  Q  e.  A ) )
110109rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. x  e.  V  ( x  =/= 
.0.  /\  ( I `  Q )  =  ( N `  { x } ) )  ->  Q  e.  A )
)
11192, 110impbid 183 . 2  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `  Q
)  =  ( N `
 { x }
) ) ) )
112 rexdifsn 3766 . 2  |-  ( E. x  e.  ( V 
\  {  .0.  }
) ( I `  Q )  =  ( N `  { x } )  <->  E. x  e.  V  ( x  =/=  .0.  /\  ( I `
 Q )  =  ( N `  {
x } ) ) )
113111, 112syl6bbr 254 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162   {csn 3653   <.cop 3656   class class class wbr 4039    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707   ` cfv 5271   iota_crio 6313   Basecbs 13164   lecple 13231   occoc 13232   0gc0g 13416   0.cp0 14159   LModclmod 15643   LSpanclspn 15744  LSAtomsclsa 29786   OPcops 29984   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DVecHcdvh 31890   DIsoHcdih 32040
This theorem is referenced by:  dihatexv2  32151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566  df-edring 31568  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041
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