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Theorem diclspsn 30634
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 2527 . . 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 4800 . . . . 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 2258 . . . . . . 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 30619 . . . . . 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 4761 . . . . 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 3233 . . . . . 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 4782 . . . . . 6  |-  Rel  ( T  X.  ( ( TEndo `  K ) `  W
) )
16 relss 4763 . . . . . 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 2766 . . . . . . 7  |-  g  e. 
_V
21 vex 2766 . . . . . . 7  |-  s  e. 
_V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 30621 . . . . . 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 29458 . . . . . . . . . . . . . . 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 30018 . . . . . . . . . . . . . 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 30206 . . . . . . . . . . . 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 2332 . . . . . . . . . 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 2258 . . . . . . . . . . . . . 14  |-  (Scalar `  U )  =  (Scalar `  U )
43 eqid 2258 . . . . . . . . . . . . . 14  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
445, 8, 41, 42, 43dvhbase 30523 . . . . . . . . . . . . 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 2718 . . . . . . . . . . 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 30208 . . . . . . . . . . . . . . . . . 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 2258 . . . . . . . . . . . . . . . . . 18  |-  ( .s
`  U )  =  ( .s `  U
)
535, 7, 8, 41, 52dvhopvsca 30542 . . . . . . . . . . . . . . . . 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 2269 . . . . . . . . . . . . . . 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 4217 . . . . . . . . . . . . . . 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 30210 . . . . . . . . . . . . . . . . . 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 2269 . . . . . . . . . . . . . . . 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 2535 . . . . . . . . . . 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 5457 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
x `  F )  =  ( s `  F ) )
7170eqeq2d 2269 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
g  =  ( x `
 F )  <->  g  =  ( s `  F
) ) )
7271ceqsrexv 2876 . . . . . . . . . . . 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 2264 . . . . . . . . . . 11  |-  ( v  =  <. g ,  s
>.  ->  ( v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  <->  <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )
) )
8180rexbidv 2539 . . . . . . . . . 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 4713 . . . . . . . . 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 2322 . . . . . . . 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 4243 . . . . . . . 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 4774 . . . 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 2325 . . . . . . . . 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 4709 . . . . . . . . . 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 30543 . . . . . . . 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 2327 . . . . . . 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 2642 . . . . 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 2372 . . 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 2316 . 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 30550 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  U  e.  LMod )
106 eqid 2258 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
1075, 7, 8, 41, 106dvhelvbasei 30528 . . . 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 15722 . . 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 2293 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 2244   E.wrex 2519   {crab 2522    C_ wss 3127   {csn 3614   <.cop 3617   class class class wbr 3997   {copab 4050    _I cid 4276    X. cxp 4659    |` cres 4663    o. ccom 4665   Rel wrel 4666   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111  Scalarcsca 13174   .scvsca 13175   lecple 13178   occoc 13179   LModclmod 15590   LSpanclspn 15691   Atomscatm 28703   HLchlt 28790   LHypclh 29423   LTrncltrn 29540   TEndoctendo 30191   DVecHcdvh 30518   DIsoCcdic 30612
This theorem is referenced by:  cdlemn5pre  30640  dih1dimc  30682
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9934  df-z 9993  df-uz 10199  df-fz 10750  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-sca 13187  df-vsca 13188  df-0g 13367  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-mnd 14330  df-grp 14452  df-minusg 14453  df-sbg 14454  df-mgp 15289  df-ring 15303  df-ur 15305  df-oppr 15368  df-dvdsr 15386  df-unit 15387  df-invr 15417  df-dvr 15428  df-drng 15477  df-lmod 15592  df-lss 15653  df-lsp 15692  df-lvec 15819  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598  df-tendo 30194  df-edring 30196  df-dvech 30519  df-dic 30613
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