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Theorem diclspsn 36009
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 2823 . . 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 5129 . . . . 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 2467 . . . . . . 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 35994 . . . . . 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 5087 . . . . 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 3585 . . . . . 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 5110 . . . . . 6  |-  Rel  ( T  X.  ( ( TEndo `  K ) `  W
) )
16 relss 5090 . . . . . 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 3116 . . . . . . 7  |-  g  e. 
_V
21 vex 3116 . . . . . . 7  |-  s  e. 
_V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 35996 . . . . . 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 34833 . . . . . . . . . . . . . . 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 35393 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3126, 28, 29, 30syl3anc 1228 . . . . . . . . . . . . 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 35581 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
s `  F )  e.  T )
3424, 25, 32, 33syl3anc 1228 . . . . . . . . . . 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 2555 . . . . . . . . . 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 1176 . . . . . . . . 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 1004 . . . . . . . . . 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 1003 . . . . . . . . . 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 830 . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  (Scalar `  U )  =  (Scalar `  U )
43 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
445, 8, 41, 42, 43dvhbase 35898 . . . . . . . . . . . . 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 3065 . . . . . . . . . . 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 35583 . . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . . . 18  |-  ( .s
`  U )  =  ( .s `  U
)
535, 7, 8, 41, 52dvhopvsca 35917 . . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . 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 4721 . . . . . . . . . . . . . . 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 35585 . . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . 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 1743 . . . . . . . . . . . . . . . 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 2970 . . . . . . . . . . 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 1305 . . . . . . . . 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 5865 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
x `  F )  =  ( s `  F ) )
7170eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
g  =  ( x `
 F )  <->  g  =  ( s `  F
) ) )
7271ceqsrexv 3237 . . . . . . . . . . . 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 977 . . . . . . . . . 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 977 . . . . . . . . . 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 2471 . . . . . . . . . . 11  |-  ( v  =  <. g ,  s
>.  ->  ( v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  <->  <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )
) )
8180rexbidv 2973 . . . . . . . . . 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 5036 . . . . . . . . 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 2545 . . . . . . . 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 4754 . . . . . . . 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 5102 . . . 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 1227 . . 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 2537 . . . . . . . . 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 5031 . . . . . . . . . 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 35918 . . . . . . . 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 1226 . . . . . . 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 2550 . . . . . . 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 2955 . . . . 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 2603 . . 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 2534 . 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 35925 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  U  e.  LMod )
106 eqid 2467 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
1075, 7, 8, 41, 106dvhelvbasei 35903 . . . 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 1226 . . 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 17448 . . 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 2511 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 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   {crab 2818    C_ wss 3476   {csn 4027   <.cop 4033   class class class wbr 4447   {copab 4504    _I cid 4790    X. cxp 4997    |` cres 5001    o. ccom 5003   Rel wrel 5004   ` cfv 5588   iota_crio 6244  (class class class)co 6284   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   lecple 14562   occoc 14563   LModclmod 17312   LSpanclspn 17417   Atomscatm 34078   HLchlt 34165   LHypclh 34798   LTrncltrn 34915   TEndoctendo 35566   DVecHcdvh 35893   DIsoCcdic 35987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-riotaBAD 33774
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-undef 7002  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-0g 14697  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-drng 17198  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lvec 17549  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-llines 34312  df-lplanes 34313  df-lvols 34314  df-lines 34315  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-lhyp 34802  df-laut 34803  df-ldil 34918  df-ltrn 34919  df-trl 34973  df-tendo 35569  df-edring 35571  df-dvech 35894  df-dic 35988
This theorem is referenced by:  cdlemn5pre  36015  dih1dimc  36057
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