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Theorem diclspsn 34836
Description: The value of isomorphism C is spanned by vector  F. Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
diclspsn.l  |-  .<_  =  ( le `  K )
diclspsn.a  |-  A  =  ( Atoms `  K )
diclspsn.h  |-  H  =  ( LHyp `  K
)
diclspsn.p  |-  P  =  ( ( oc `  K ) `  W
)
diclspsn.t  |-  T  =  ( ( LTrn `  K
) `  W )
diclspsn.i  |-  I  =  ( ( DIsoC `  K
) `  W )
diclspsn.u  |-  U  =  ( ( DVecH `  K
) `  W )
diclspsn.n  |-  N  =  ( LSpan `  U )
diclspsn.f  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
Assertion
Ref Expression
diclspsn  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( N `
 { <. F , 
(  _I  |`  T )
>. } ) )
Distinct variable groups:    .<_ , f    P, f    A, f    f, H    T, f    f, K    Q, f    f, W
Allowed substitution hints:    U( f)    F( f)    I( f)    N( f)

Proof of Theorem diclspsn
Dummy variables  g 
s  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2722 . . 3  |-  { v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) }  =  { v  |  ( v  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  /\  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) }
2 relopab 4964 . . . . 5  |-  Rel  { <. y ,  z >.  |  ( y  =  ( z `  F
)  /\  z  e.  ( ( TEndo `  K
) `  W )
) }
3 diclspsn.l . . . . . . 7  |-  .<_  =  ( le `  K )
4 diclspsn.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 diclspsn.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
6 diclspsn.p . . . . . . 7  |-  P  =  ( ( oc `  K ) `  W
)
7 diclspsn.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
8 eqid 2441 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
9 diclspsn.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
10 diclspsn.f . . . . . . 7  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
113, 4, 5, 6, 7, 8, 9, 10dicval2 34821 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. y ,  z >.  |  ( y  =  ( z `
 F )  /\  z  e.  ( ( TEndo `  K ) `  W ) ) } )
1211releqd 4922 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Rel  ( I `  Q )  <->  Rel  { <. y ,  z >.  |  ( y  =  ( z `
 F )  /\  z  e.  ( ( TEndo `  K ) `  W ) ) } ) )
132, 12mpbiri 233 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Rel  ( I `  Q
) )
14 ssrab2 3435 . . . . . 6  |-  { v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) }  C_  ( T  X.  ( ( TEndo `  K
) `  W )
)
15 relxp 4945 . . . . . 6  |-  Rel  ( T  X.  ( ( TEndo `  K ) `  W
) )
16 relss 4925 . . . . . 6  |-  ( { v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  |  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) }  C_  ( T  X.  ( ( TEndo `  K ) `  W
) )  ->  ( Rel  ( T  X.  (
( TEndo `  K ) `  W ) )  ->  Rel  { v  e.  ( T  X.  ( (
TEndo `  K ) `  W ) )  |  E. x  e.  (
Base `  (Scalar `  U
) ) v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. ) } ) )
1714, 15, 16mp2 9 . . . . 5  |-  Rel  {
v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  |  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) }
1817a1i 11 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Rel  { v  e.  ( T  X.  ( (
TEndo `  K ) `  W ) )  |  E. x  e.  (
Base `  (Scalar `  U
) ) v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. ) } )
19 id 22 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
20 vex 2973 . . . . . . 7  |-  g  e. 
_V
21 vex 2973 . . . . . . 7  |-  s  e. 
_V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 34823 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. g ,  s
>.  e.  ( I `  Q )  <->  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) ) )
23 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  g  =  ( s `  F ) )
24 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
25 simprr 756 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  s  e.  ( ( TEndo `  K
) `  W )
)
26 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
273, 4, 5, 6lhpocnel2 33660 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2827adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
29 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
303, 4, 5, 7, 10ltrniotacl 34220 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3126, 28, 29, 30syl3anc 1218 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3231adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  F  e.  T )
335, 7, 8tendocl 34408 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
s `  F )  e.  T )
3424, 25, 32, 33syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  (
s `  F )  e.  T )
3523, 34eqeltrd 2515 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  g  e.  T )
3635, 25, 233jca 1168 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  =  ( s `  F )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  (
g  e.  T  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  g  =  ( s `  F ) ) )
37 simpr3 996 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  g  =  (
s `  F )
) )  ->  g  =  ( s `  F ) )
38 simpr2 995 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  g  =  (
s `  F )
) )  ->  s  e.  ( ( TEndo `  K
) `  W )
)
3937, 38jca 532 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  g  =  (
s `  F )
) )  ->  (
g  =  ( s `
 F )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) )
4036, 39impbida 828 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( g  =  ( s `  F
)  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  g  =  (
s `  F )
) ) )
41 diclspsn.u . . . . . . . . . . . . . 14  |-  U  =  ( ( DVecH `  K
) `  W )
42 eqid 2441 . . . . . . . . . . . . . 14  |-  (Scalar `  U )  =  (Scalar `  U )
43 eqid 2441 . . . . . . . . . . . . . 14  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
445, 8, 41, 42, 43dvhbase 34725 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
4544adantr 465 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Base `  (Scalar `  U
) )  =  ( ( TEndo `  K ) `  W ) )
4645rexeqdv 2922 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( E. x  e.  ( Base `  (Scalar `  U ) ) <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  <->  E. x  e.  ( (
TEndo `  K ) `  W ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) )
47 simpll 753 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
48 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  x  e.  ( ( TEndo `  K
) `  W )
)
4931adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  F  e.  T )
505, 7, 8tendoidcl 34410 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
5150ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W ) )
52 eqid 2441 . . . . . . . . . . . . . . . . . 18  |-  ( .s
`  U )  =  ( .s `  U
)
535, 7, 8, 41, 52dvhopvsca 34744 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  F  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) ) )  -> 
( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  =  <. ( x `  F ) ,  ( x  o.  (  _I  |`  T ) ) >.
)
5447, 48, 49, 51, 53syl13anc 1220 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( x
( .s `  U
) <. F ,  (  _I  |`  T ) >. )  =  <. (
x `  F ) ,  ( x  o.  (  _I  |`  T ) ) >. )
5554eqeq2d 2452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. )  <->  <. g ,  s
>.  =  <. ( x `
 F ) ,  ( x  o.  (  _I  |`  T ) )
>. ) )
5620, 21opth 4564 . . . . . . . . . . . . . . 15  |-  ( <.
g ,  s >.  =  <. ( x `  F ) ,  ( x  o.  (  _I  |`  T ) ) >.  <->  ( g  =  ( x `
 F )  /\  s  =  ( x  o.  (  _I  |`  T ) ) ) )
5755, 56syl6bb 261 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. )  <->  ( g  =  ( x `  F
)  /\  s  =  ( x  o.  (  _I  |`  T ) ) ) ) )
585, 7, 8tendo1mulr 34412 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W ) )  -> 
( x  o.  (  _I  |`  T ) )  =  x )
5958adantlr 714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( x  o.  (  _I  |`  T ) )  =  x )
6059eqeq2d 2452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( s  =  ( x  o.  (  _I  |`  T ) )  <->  s  =  x ) )
61 equcom 1732 . . . . . . . . . . . . . . . 16  |-  ( s  =  x  <->  x  =  s )
6260, 61syl6bb 261 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( s  =  ( x  o.  (  _I  |`  T ) )  <->  x  =  s
) )
6362anbi2d 703 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( (
g  =  ( x `
 F )  /\  s  =  ( x  o.  (  _I  |`  T ) ) )  <->  ( g  =  ( x `  F )  /\  x  =  s ) ) )
6457, 63bitrd 253 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. )  <->  ( g  =  ( x `  F
)  /\  x  =  s ) ) )
65 ancom 450 . . . . . . . . . . . . 13  |-  ( ( g  =  ( x `
 F )  /\  x  =  s )  <->  ( x  =  s  /\  g  =  ( x `  F ) ) )
6664, 65syl6bb 261 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. )  <->  ( x  =  s  /\  g  =  ( x `  F
) ) ) )
6766rexbidva 2730 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( E. x  e.  ( ( TEndo `  K
) `  W ) <. g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  <->  E. x  e.  ( (
TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) ) )
6846, 67bitrd 253 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( E. x  e.  ( Base `  (Scalar `  U ) ) <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  <->  E. x  e.  ( (
TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) ) )
69683anbi3d 1295 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  (
Base `  (Scalar `  U
) ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) )  <->  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) ) ) )
70 fveq1 5688 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
x `  F )  =  ( s `  F ) )
7170eqeq2d 2452 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
g  =  ( x `
 F )  <->  g  =  ( s `  F
) ) )
7271ceqsrexv 3091 . . . . . . . . . . . 12  |-  ( s  e.  ( ( TEndo `  K ) `  W
)  ->  ( E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) )  <->  g  =  ( s `  F
) ) )
7372pm5.32i 637 . . . . . . . . . . 11  |-  ( ( s  e.  ( (
TEndo `  K ) `  W )  /\  E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) )  <-> 
( s  e.  ( ( TEndo `  K ) `  W )  /\  g  =  ( s `  F ) ) )
7473anbi2i 694 . . . . . . . . . 10  |-  ( ( g  e.  T  /\  ( s  e.  ( ( TEndo `  K ) `  W )  /\  E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) ) )  <->  ( g  e.  T  /\  ( s  e.  ( ( TEndo `  K ) `  W
)  /\  g  =  ( s `  F
) ) ) )
75 3anass 969 . . . . . . . . . 10  |-  ( ( g  e.  T  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) )  <-> 
( g  e.  T  /\  ( s  e.  ( ( TEndo `  K ) `  W )  /\  E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) ) ) )
76 3anass 969 . . . . . . . . . 10  |-  ( ( g  e.  T  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  g  =  ( s `  F ) )  <->  ( g  e.  T  /\  (
s  e.  ( (
TEndo `  K ) `  W )  /\  g  =  ( s `  F ) ) ) )
7774, 75, 763bitr4i 277 . . . . . . . . 9  |-  ( ( g  e.  T  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) ) )  <-> 
( g  e.  T  /\  s  e.  (
( TEndo `  K ) `  W )  /\  g  =  ( s `  F ) ) )
7869, 77syl6rbb 262 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  g  =  (
s `  F )
)  <->  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  (
Base `  (Scalar `  U
) ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) ) )
7940, 78bitrd 253 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( g  =  ( s `  F
)  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  (
Base `  (Scalar `  U
) ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) ) )
80 eqeq1 2447 . . . . . . . . . . 11  |-  ( v  =  <. g ,  s
>.  ->  ( v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  <->  <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )
) )
8180rexbidv 2734 . . . . . . . . . 10  |-  ( v  =  <. g ,  s
>.  ->  ( E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  <->  E. x  e.  ( Base `  (Scalar `  U )
) <. g ,  s
>.  =  ( x
( .s `  U
) <. F ,  (  _I  |`  T ) >. ) ) )
8281rabxp 4873 . . . . . . . . 9  |-  { v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) }  =  { <. g ,  s >.  |  ( g  e.  T  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  E. x  e.  ( Base `  (Scalar `  U )
) <. g ,  s
>.  =  ( x
( .s `  U
) <. F ,  (  _I  |`  T ) >. ) ) }
8382eleq2i 2505 . . . . . . . 8  |-  ( <.
g ,  s >.  e.  { v  e.  ( T  X.  ( (
TEndo `  K ) `  W ) )  |  E. x  e.  (
Base `  (Scalar `  U
) ) v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. ) } 
<-> 
<. g ,  s >.  e.  { <. g ,  s
>.  |  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  (
Base `  (Scalar `  U
) ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) } )
84 opabid 4594 . . . . . . . 8  |-  ( <.
g ,  s >.  e.  { <. g ,  s
>.  |  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  (
Base `  (Scalar `  U
) ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) }  <->  ( g  e.  T  /\  s  e.  ( ( TEndo `  K
) `  W )  /\  E. x  e.  (
Base `  (Scalar `  U
) ) <. g ,  s >.  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) )
8583, 84bitr2i 250 . . . . . . 7  |-  ( ( g  e.  T  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  E. x  e.  ( Base `  (Scalar `  U )
) <. g ,  s
>.  =  ( x
( .s `  U
) <. F ,  (  _I  |`  T ) >. ) )  <->  <. g ,  s >.  e.  { v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) } )
8679, 85syl6bb 261 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( g  =  ( s `  F
)  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  <. g ,  s
>.  e.  { v  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  |  E. x  e.  (
Base `  (Scalar `  U
) ) v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. ) } ) )
8722, 86bitrd 253 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. g ,  s
>.  e.  ( I `  Q )  <->  <. g ,  s >.  e.  { v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) } ) )
8887eqrelrdv2 4937 . . . 4  |-  ( ( ( Rel  ( I `
 Q )  /\  Rel  { v  e.  ( T  X.  ( (
TEndo `  K ) `  W ) )  |  E. x  e.  (
Base `  (Scalar `  U
) ) v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. ) } )  /\  (
( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( I `  Q )  =  {
v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  |  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) } )
8913, 18, 19, 88syl21anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) } )
90 simpll 753 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9145eleq2d 2508 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( x  e.  (
Base `  (Scalar `  U
) )  <->  x  e.  ( ( TEndo `  K
) `  W )
) )
9291biimpa 484 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  x  e.  ( ( TEndo `  K ) `  W ) )
9350adantr 465 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
(  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
94 opelxpi 4869 . . . . . . . . . 10  |-  ( ( F  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )  ->  <. F , 
(  _I  |`  T )
>.  e.  ( T  X.  ( ( TEndo `  K
) `  W )
) )
9531, 93, 94syl2anc 661 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  <. F ,  (  _I  |`  T ) >.  e.  ( T  X.  ( (
TEndo `  K ) `  W ) ) )
9695adantr 465 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  <. F ,  (  _I  |`  T ) >.  e.  ( T  X.  ( ( TEndo `  K
) `  W )
) )
975, 7, 8, 41, 52dvhvscacl 34745 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  <. F ,  (  _I  |`  T ) >.  e.  ( T  X.  ( ( TEndo `  K
) `  W )
) ) )  -> 
( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) )
9890, 92, 96, 97syl12anc 1216 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) )
99 eleq1a 2510 . . . . . . 7  |-  ( ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. )  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  ->  (
v  =  ( x ( .s `  U
) <. F ,  (  _I  |`  T ) >. )  ->  v  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) ) )
10098, 99syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  ( v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  ->  v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) ) ) )
101100rexlimdva 2839 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  ->  v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) ) ) )
102101pm4.71rd 635 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )  <->  ( v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  /\  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) ) ) )
103102abbidv 2555 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  { v  |  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) }  =  {
v  |  ( v  e.  ( T  X.  ( ( TEndo `  K
) `  W )
)  /\  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )
) } )
1041, 89, 1033eqtr4a 2499 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { v  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) } )
1055, 41, 26dvhlmod 34752 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  U  e.  LMod )
106 eqid 2441 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
1075, 7, 8, 41, 106dvhelvbasei 34730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W ) ) )  ->  <. F ,  (  _I  |`  T ) >.  e.  ( Base `  U
) )
10826, 31, 93, 107syl12anc 1216 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  <. F ,  (  _I  |`  T ) >.  e.  (
Base `  U )
)
109 diclspsn.n . . . 4  |-  N  =  ( LSpan `  U )
11042, 43, 106, 52, 109lspsn 17081 . . 3  |-  ( ( U  e.  LMod  /\  <. F ,  (  _I  |`  T )
>.  e.  ( Base `  U
) )  ->  ( N `  { <. F , 
(  _I  |`  T )
>. } )  =  {
v  |  E. x  e.  ( Base `  (Scalar `  U ) ) v  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. ) } )
111105, 108, 110syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( N `  { <. F ,  (  _I  |`  T ) >. } )  =  { v  |  E. x  e.  (
Base `  (Scalar `  U
) ) v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. ) } )
112104, 111eqtr4d 2476 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( N `
 { <. F , 
(  _I  |`  T )
>. } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2427   E.wrex 2714   {crab 2717    C_ wss 3326   {csn 3875   <.cop 3881   class class class wbr 4290   {copab 4347    _I cid 4629    X. cxp 4836    |` cres 4840    o. ccom 4842   Rel wrel 4843   ` cfv 5416   iota_crio 6049  (class class class)co 6089   Basecbs 14172  Scalarcsca 14239   .scvsca 14240   lecple 14243   occoc 14244   LModclmod 16946   LSpanclspn 17050   Atomscatm 32905   HLchlt 32992   LHypclh 33625   LTrncltrn 33742   TEndoctendo 34393   DVecHcdvh 34720   DIsoCcdic 34814
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-riotaBAD 32601
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-tpos 6743  df-undef 6790  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-0g 14378  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mgp 16590  df-ur 16602  df-rng 16645  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-dvr 16773  df-drng 16832  df-lmod 16948  df-lss 17012  df-lsp 17051  df-lvec 17182  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-llines 33139  df-lplanes 33140  df-lvols 33141  df-lines 33142  df-psubsp 33144  df-pmap 33145  df-padd 33437  df-lhyp 33629  df-laut 33630  df-ldil 33745  df-ltrn 33746  df-trl 33800  df-tendo 34396  df-edring 34398  df-dvech 34721  df-dic 34815
This theorem is referenced by:  cdlemn5pre  34842  dih1dimc  34884
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