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Theorem diclspsn 30185
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
StepHypRef Expression
1 df-rab 2516 . . 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 4719 . . . . 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 2253 . . . . . . 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 30170 . . . . . 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 4680 . . . . 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 226 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Rel  ( I `  Q
) )
14 ssrab2 3179 . . . . . 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 4701 . . . . . 6  |-  Rel  ( T  X.  ( ( TEndo `  K ) `  W
) )
16 relss 4682 . . . . . 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 19 . . . . 5  |-  Rel  {
v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  |  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) }
1817a1i 12 . . . 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 21 . . . 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 2730 . . . . . . 7  |-  g  e. 
_V
21 vex 2730 . . . . . . 7  |-  s  e. 
_V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 30172 . . . . . 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 735 . . . . . . . . . . 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 733 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 445 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
273, 4, 5, 6lhpocnel2 29009 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2827adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
29 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
303, 4, 5, 7, 10ltrniotacl 29569 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3126, 28, 29, 30syl3anc 1187 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3231adantr 453 . . . . . . . . . . . 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 29757 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
s `  F )  e.  T )
3424, 25, 32, 33syl3anc 1187 . . . . . . . . . . 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 2327 . . . . . . . . . 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 1137 . . . . . . . . 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 968 . . . . . . . . . 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 967 . . . . . . . . . 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 520 . . . . . . . . 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 808 . . . . . . . 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 2253 . . . . . . . . . . . . . 14  |-  (Scalar `  U )  =  (Scalar `  U )
43 eqid 2253 . . . . . . . . . . . . . 14  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
445, 8, 41, 42, 43dvhbase 30074 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
4544adantr 453 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Base `  (Scalar `  U
) )  =  ( ( TEndo `  K ) `  W ) )
4645rexeqdv 2695 . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  x  e.  ( ( TEndo `  K
) `  W )
)
4931adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  F  e.  T )
505, 7, 8tendoidcl 29759 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
5150ad2antrr 709 . . . . . . . . . . . . . . . . 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 2253 . . . . . . . . . . . . . . . . . 18  |-  ( .s
`  U )  =  ( .s `  U
)
535, 7, 8, 41, 52dvhopvsca 30093 . . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . . 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 2264 . . . . . . . . . . . . . . 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 4138 . . . . . . . . . . . . . . 15  |-  ( <.
g ,  s >.  =  <. ( x `  F ) ,  ( x  o.  (  _I  |`  T ) ) >.  <->  ( g  =  ( x `
 F )  /\  s  =  ( x  o.  (  _I  |`  T ) ) ) )
5755, 56syl6bb 254 . . . . . . . . . . . . . 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 29761 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W ) )  -> 
( x  o.  (  _I  |`  T ) )  =  x )
5958adantlr 698 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( x  o.  (  _I  |`  T ) )  =  x )
6059eqeq2d 2264 . . . . . . . . . . . . . . . 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 1824 . . . . . . . . . . . . . . . 16  |-  ( s  =  x  <->  x  =  s )
6260, 61syl6bb 254 . . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . 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 439 . . . . . . . . . . . . 13  |-  ( ( g  =  ( x `
 F )  /\  x  =  s )  <->  ( x  =  s  /\  g  =  ( x `  F ) ) )
6664, 65syl6bb 254 . . . . . . . . . . . 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 2524 . . . . . . . . . . 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 246 . . . . . . . . . 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 1263 . . . . . . . . 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 5376 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
x `  F )  =  ( s `  F ) )
7170eqeq2d 2264 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
g  =  ( x `
 F )  <->  g  =  ( s `  F
) ) )
7271ceqsrexv 2838 . . . . . . . . . . . 12  |-  ( s  e.  ( ( TEndo `  K ) `  W
)  ->  ( E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) )  <->  g  =  ( s `  F
) ) )
7372pm5.32i 621 . . . . . . . . . . 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 678 . . . . . . . . . 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 943 . . . . . . . . . 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 943 . . . . . . . . . 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 270 . . . . . . . . 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 255 . . . . . . . 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 246 . . . . . . 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 2259 . . . . . . . . . . 11  |-  ( v  =  <. g ,  s
>.  ->  ( v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  <->  <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )
) )
8180rexbidv 2528 . . . . . . . . . 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 4632 . . . . . . . . 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 2317 . . . . . . . 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 4164 . . . . . . . 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 243 . . . . . . 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 254 . . . . . 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 246 . . . . 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 4693 . . . 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 1186 . . 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 733 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9145eleq2d 2320 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( x  e.  (
Base `  (Scalar `  U
) )  <->  x  e.  ( ( TEndo `  K
) `  W )
) )
9291biimpa 472 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  x  e.  ( ( TEndo `  K ) `  W ) )
9350adantr 453 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
(  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
94 opelxpi 4628 . . . . . . . . . 10  |-  ( ( F  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )  ->  <. F , 
(  _I  |`  T )
>.  e.  ( T  X.  ( ( TEndo `  K
) `  W )
) )
9531, 93, 94syl2anc 645 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  <. F ,  (  _I  |`  T ) >.  e.  ( T  X.  ( (
TEndo `  K ) `  W ) ) )
9695adantr 453 . . . . . . . 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 30094 . . . . . . . 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 1185 . . . . . . 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 2322 . . . . . . 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 17 . . . . . 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 2629 . . . . 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 619 . . . 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 2363 . . 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 2311 . 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 30101 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  U  e.  LMod )
106 eqid 2253 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
1075, 7, 8, 41, 106dvhelvbasei 30079 . . . 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 1185 . . 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 15594 . . 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 645 . 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 2288 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( N `
 { <. F , 
(  _I  |`  T )
>. } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   {cab 2239   E.wrex 2510   {crab 2512    C_ wss 3078   {csn 3544   <.cop 3547   class class class wbr 3920   {copab 3973    _I cid 4197    X. cxp 4578    |` cres 4582    o. ccom 4584   Rel wrel 4585   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022  Scalarcsca 13085   .scvsca 13086   lecple 13089   occoc 13090   LModclmod 15462   LSpanclspn 15563   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   TEndoctendo 29742   DVecHcdvh 30069   DIsoCcdic 30163
This theorem is referenced by:  cdlemn5pre  30191  dih1dimc  30233
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-grp 14324  df-minusg 14325  df-sbg 14326  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149  df-tendo 29745  df-edring 29747  df-dvech 30070  df-dic 30164
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