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Theorem lfl1dim 29608
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 2679 . 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 16137 . . . . . . . . . . . 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 2408 . . . . . . . . . . . 12  |-  ( 0g
`  D )  =  ( 0g `  D
)
85, 6, 7lmod0cl 15935 . . . . . . . . . . 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 29569 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) )  =  ( V  X.  { ( 0g `  D ) } ) )
1911, 18eqtr4d 2443 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  F )  /\  g  =  ( V  X.  { ( 0g `  D ) } ) )  ->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) )
20 sneq 3789 . . . . . . . . . . . . 13  |-  ( k  =  ( 0g `  D )  ->  { k }  =  { ( 0g `  D ) } )
2120xpeq2d 4865 . . . . . . . . . . . 12  |-  ( k  =  ( 0g `  D )  ->  ( V  X.  { k } )  =  ( V  X.  { ( 0g
`  D ) } ) )
2221oveq2d 6060 . . . . . . . . . . 11  |-  ( k  =  ( 0g `  D )  ->  ( G  o F  .x.  ( V  X.  { k } ) )  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2322eqeq2d 2419 . . . . . . . . . 10  |-  ( k  =  ( 0g `  D )  ->  (
g  =  ( G  o F  .x.  ( V  X.  { k } ) )  <->  g  =  ( G  o F  .x.  ( V  X.  {
( 0g `  D
) } ) ) ) )
2423rspcev 3016 . . . . . . . . 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 29583 . . . . . . . . . . . 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 29581 . . . . . . . . . . . . . . . 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 3339 . . . . . . . . . . . . 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 3329 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  g  e.  F )  /\  G  =  ( V  X.  { ( 0g
`  D ) } ) )  /\  ( L `  G )  C_  ( L `  g
) )  ->  ( L `  g )  =  V )
405, 7, 12, 13, 28lkr0f 29581 . . . . . . . . . . . 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 29569 . . . . . . . . . 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 2443 . . . . . . . . 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 2408 . . . . . . . . 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 29592 . . . . . . . . . 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 29592 . . . . . . . . . 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 29475 . . . . . . . 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 29587 . . . . . . . . . 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 2650 . . . . . 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 29585 . . . . . . . . 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 5691 . . . . . . . . . 10  |-  ( g  =  ( G  o F  .x.  ( V  X.  { k } ) )  ->  ( L `  g )  =  ( L `  ( G  o F  .x.  ( V  X.  { k } ) ) ) )
7473sseq2d 3340 . . . . . . . . 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 2793 . . . . . 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 29564 . . . . . . . 8  |-  ( (
ph  /\  k  e.  K )  ->  ( G  o F  .x.  ( V  X.  { k } ) )  e.  F
)
84 eleq1a 2477 . . . . . . . 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 2687 . . . . 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 2826 . . . . 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 2522 . 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 2452 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 2394    =/= wne 2571   E.wrex 2671   {crab 2674    C_ wss 3284   {csn 3778    X. cxp 4839   ` cfv 5417  (class class class)co 6044    o Fcof 6266   Basecbs 13428   .rcmulr 13489  Scalarcsca 13491   0gc0g 13682   LModclmod 15909   LVecclvec 16133  LSHypclsh 29462  LFnlclfn 29544  LKerclk 29572
This theorem is referenced by:  ldual1dim  29653
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-0g 13686  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lshyp 29464  df-lfl 29545  df-lkr 29573
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