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Theorem lfl1dim 29604
Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
Hypotheses
Ref Expression
lfl1dim.v  |-  V  =  ( Base `  W
)
lfl1dim.d  |-  D  =  (Scalar `  W )
lfl1dim.f  |-  F  =  (LFnl `  W )
lfl1dim.l  |-  L  =  (LKer `  W )
lfl1dim.k  |-  K  =  ( Base `  D
)
lfl1dim.t  |-  .x.  =  ( .r `  D )
lfl1dim.w  |-  ( ph  ->  W  e.  LVec )
lfl1dim.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lfl1dim  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) } )
Distinct variable groups:    D, k    k, F    k, G    k, K    k, L    k, V    k, W    g, k, ph    .x. , k
Allowed substitution hints:    D( g)    .x. ( g)    F( g)    G( g)    K( g)    L( g)    V( g)    W( g)

Proof of Theorem lfl1dim
StepHypRef Expression
1 df-rab 2675 . 2  |-  { g  e.  F  |  ( L `  G ) 
C_  ( L `  g ) }  =  { g  |  ( g  e.  F  /\  ( L `  G ) 
C_  ( L `  g ) ) }
2 lfl1dim.w . . . . . . . . . . . 12  |-  ( ph  ->  W  e.  LVec )
3 lveclmod 16133 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  W  e. 
LMod )
42, 3syl 16 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LMod )
5 lfl1dim.d . . . . . . . . . . . 12  |-  D  =  (Scalar `  W )
6 lfl1dim.k . . . . . . . . . . . 12  |-  K  =  ( Base `  D
)
7 eqid 2404 . . . . . . . . . . . 12  |-  ( 0g
`  D )  =  ( 0g `  D
)
85, 6, 7lmod0cl 15931 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  ( 0g
`  D )  e.  K )
94, 8syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  D
)  e.  K )
109ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( 0g `  D )  e.  K
)
11 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( V  X.  { ( 0g `  D ) } ) )
12 lfl1dim.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
13 lfl1dim.f . . . . . . . . . . 11  |-  F  =  (LFnl `  W )
14 lfl1dim.t . . . . . . . . . . 11  |-  .x.  =  ( .r `  D )
154ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  W  e.  LMod )
16 lfl1dim.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  F )
1716ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  G  e.  F )
1812, 5, 13, 6, 14, 7, 15, 17lfl0sc 29565 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
1911, 18eqtr4d 2439 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) )
20 sneq 3785 . . . . . . . . . . . . 13  |-  ( k  =  ( 0g `  D )  ->  { k }  =  { ( 0g `  D ) } )
2120xpeq2d 4861 . . . . . . . . . . . 12  |-  ( k  =  ( 0g `  D )  ->  ( V  X.  { k } )  =  ( V  X.  { ( 0g
`  D ) } ) )
2221oveq2d 6056 . . . . . . . . . . 11  |-  ( k  =  ( 0g `  D )  ->  ( G  o F  .x.  ( V  X.  { k } ) )  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2322eqeq2d 2415 . . . . . . . . . 10  |-  ( k  =  ( 0g `  D )  ->  (
g  =  ( G  o F  .x.  ( V  X.  { k } ) )  <->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) ) )
2423rspcev 3012 . . . . . . . . 9  |-  ( ( ( 0g `  D
)  e.  K  /\  g  =  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) )
2510, 19, 24syl2anc 643 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
2625a1d 23 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
279ad3antrrr 711 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( 0g `  D )  e.  K )
28 lfl1dim.l . . . . . . . . . . . . 13  |-  L  =  (LKer `  W )
294ad3antrrr 711 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  W  e.  LMod )
30 simpllr 736 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  e.  F )
3112, 13, 28, 29, 30lkrssv 29579 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  C_  V )
324adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  g  e.  F )  ->  W  e.  LMod )
3316adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  g  e.  F )  ->  G  e.  F )
345, 7, 12, 13, 28lkr0f 29577 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3532, 33, 34syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  =  V  <->  G  =  ( V  X.  { ( 0g `  D ) } ) ) )
3635biimpar 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( L `  G )  =  V )
3736sseq1d 3335 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  V  C_  ( L `
 g ) ) )
3837biimpa 471 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  V  C_  ( L `  g
) )
3931, 38eqssd 3325 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  =  V )
405, 7, 12, 13, 28lkr0f 29577 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  g  e.  F )  ->  (
( L `  g
)  =  V  <->  g  =  ( V  X.  { ( 0g `  D ) } ) ) )
4129, 30, 40syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  (
( L `  g
)  =  V  <->  g  =  ( V  X.  { ( 0g `  D ) } ) ) )
4239, 41mpbid 202 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( V  X.  { ( 0g `  D ) } ) )
4316ad3antrrr 711 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  G  e.  F )
4412, 5, 13, 6, 14, 7, 29, 43lfl0sc 29565 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
4542, 44eqtr4d 2439 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  g  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
4627, 45, 24syl2anc 643 . . . . . . . 8  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
4746ex 424 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
48 eqid 2404 . . . . . . . . 9  |-  (LSHyp `  W )  =  (LSHyp `  W )
492ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  W  e.  LVec )
5016ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  G  e.  F )
51 simprr 734 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  G  =/=  ( V  X.  { ( 0g `  D ) } ) )
5212, 5, 7, 48, 13, 28lkrshp 29588 . . . . . . . . . 10  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
5349, 50, 51, 52syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  G )  e.  (LSHyp `  W ) )
54 simplr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  g  e.  F )
55 simprl 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  g  =/=  ( V  X.  { ( 0g `  D ) } ) )
5612, 5, 7, 48, 13, 28lkrshp 29588 . . . . . . . . . 10  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {
( 0g `  D
) } ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5749, 54, 55, 56syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( L `  g )  e.  (LSHyp `  W ) )
5848, 49, 53, 57lshpcmp 29471 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  ( L `  G )  =  ( L `  g ) ) )
592ad3antrrr 711 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  W  e.  LVec )
6016ad3antrrr 711 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  G  e.  F )
61 simpllr 736 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  g  e.  F )
62 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  ( L `  G )  =  ( L `  g ) )
635, 6, 14, 12, 13, 28eqlkr2 29583 . . . . . . . . . 10  |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  g  e.  F )  /\  ( L `  G
)  =  ( L `
 g ) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) )
6459, 60, 61, 62, 63syl121anc 1189 . . . . . . . . 9  |-  ( ( ( ( ph  /\  g  e.  F )  /\  ( g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  /\  ( L `
 G )  =  ( L `  g
) )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )
6564ex 424 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( ( L `  G )  =  ( L `  g )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
6658, 65sylbid 207 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  F )  /\  (
g  =/=  ( V  X.  { ( 0g
`  D ) } )  /\  G  =/=  ( V  X.  {
( 0g `  D
) } ) ) )  ->  ( ( L `  G )  C_  ( L `  g
)  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
6726, 47, 66pm2.61da2ne 2646 . . . . . 6  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  ->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
682ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  W  e.  LVec )
6916ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  G  e.  F )
70 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  k  e.  K )
7112, 5, 6, 14, 13, 28, 68, 69, 70lkrscss 29581 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  k  e.  K )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7271ex 424 . . . . . . . 8  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { k } ) ) ) ) )
73 fveq2 5687 . . . . . . . . . 10  |-  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  g )  =  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7473sseq2d 3336 . . . . . . . . 9  |-  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( ( L `  G )  C_  ( L `  g
)  <->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) ) )
7574biimprcd 217 . . . . . . . 8  |-  ( ( L `  G ) 
C_  ( L `  ( G  o F  .x.  ( V  X.  {
k } ) ) )  ->  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g )
) )
7672, 75syl6 31 . . . . . . 7  |-  ( (
ph  /\  g  e.  F )  ->  (
k  e.  K  -> 
( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  -> 
( L `  G
)  C_  ( L `  g ) ) ) )
7776rexlimdv 2789 . . . . . 6  |-  ( (
ph  /\  g  e.  F )  ->  ( E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  G )  C_  ( L `  g
) ) )
7867, 77impbid 184 . . . . 5  |-  ( (
ph  /\  g  e.  F )  ->  (
( L `  G
)  C_  ( L `  g )  <->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
7978pm5.32da 623 . . . 4  |-  ( ph  ->  ( ( g  e.  F  /\  ( L `
 G )  C_  ( L `  g ) )  <->  ( g  e.  F  /\  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) ) )
804adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  K )  ->  W  e.  LMod )
8116adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  K )  ->  G  e.  F )
82 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  K )  ->  k  e.  K )
8312, 5, 6, 14, 13, 80, 81, 82lflvscl 29560 . . . . . . . 8  |-  ( (
ph  /\  k  e.  K )  ->  ( G  o F  .x.  ( V  X.  { k } ) )  e.  F
)
84 eleq1a 2473 . . . . . . . 8  |-  ( ( G  o F  .x.  ( V  X.  { k } ) )  e.  F  ->  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  g  e.  F ) )
8583, 84syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  K )  ->  (
g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  g  e.  F ) )
8685pm4.71rd 617 . . . . . 6  |-  ( (
ph  /\  k  e.  K )  ->  (
g  =  ( G  o F  .x.  ( V  X.  { k } ) )  <->  ( g  e.  F  /\  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) ) ) )
8786rexbidva 2683 . . . . 5  |-  ( ph  ->  ( E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) )  <->  E. k  e.  K  ( g  e.  F  /\  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) ) ) )
88 r19.42v 2822 . . . . 5  |-  ( E. k  e.  K  ( g  e.  F  /\  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) )  <->  ( g  e.  F  /\  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) ) )
8987, 88syl6rbb 254 . . . 4  |-  ( ph  ->  ( ( g  e.  F  /\  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  {
k } ) ) )  <->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
9079, 89bitrd 245 . . 3  |-  ( ph  ->  ( ( g  e.  F  /\  ( L `
 G )  C_  ( L `  g ) )  <->  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
9190abbidv 2518 . 2  |-  ( ph  ->  { g  |  ( g  e.  F  /\  ( L `  G ) 
C_  ( L `  g ) ) }  =  { g  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) } )
921, 91syl5eq 2448 1  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   E.wrex 2667   {crab 2670    C_ wss 3280   {csn 3774    X. cxp 4835   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Basecbs 13424   .rcmulr 13485  Scalarcsca 13487   0gc0g 13678   LModclmod 15905   LVecclvec 16129  LSHypclsh 29458  LFnlclfn 29540  LKerclk 29568
This theorem is referenced by:  ldual1dim  29649
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lshyp 29460  df-lfl 29541  df-lkr 29569
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