Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diclspsn Unicode version

Theorem diclspsn 31457
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 2554 . . 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 4814 . . . . 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 2285 . . . . . . 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 31442 . . . . . 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 4775 . . . . 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 224 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Rel  ( I `  Q
) )
14 ssrab2 3260 . . . . . 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 4796 . . . . . 6  |-  Rel  ( T  X.  ( ( TEndo `  K ) `  W
) )
16 relss 4777 . . . . . 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 17 . . . . 5  |-  Rel  {
v  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  |  E. x  e.  ( Base `  (Scalar `  U )
) v  =  ( x ( .s `  U ) <. F , 
(  _I  |`  T )
>. ) }
1817a1i 10 . . . 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 19 . . . 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 2793 . . . . . . 7  |-  g  e. 
_V
21 vex 2793 . . . . . . 7  |-  s  e. 
_V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 31444 . . . . . 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 732 . . . . . . . . . . 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 730 . . . . . . . . . . . 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 733 . . . . . . . . . . . 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 443 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
273, 4, 5, 6lhpocnel2 30281 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2827adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
29 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
303, 4, 5, 7, 10ltrniotacl 30841 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3126, 28, 29, 30syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
3231adantr 451 . . . . . . . . . . . 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 31029 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
s `  F )  e.  T )
3424, 25, 32, 33syl3anc 1182 . . . . . . . . . . 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 2359 . . . . . . . . . 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 1132 . . . . . . . . 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 963 . . . . . . . . . 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 962 . . . . . . . . . 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 518 . . . . . . . . 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 805 . . . . . . . 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 2285 . . . . . . . . . . . . . 14  |-  (Scalar `  U )  =  (Scalar `  U )
43 eqid 2285 . . . . . . . . . . . . . 14  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
445, 8, 41, 42, 43dvhbase 31346 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
4544adantr 451 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Base `  (Scalar `  U
) )  =  ( ( TEndo `  K ) `  W ) )
4645rexeqdv 2745 . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  x  e.  ( ( TEndo `  K
) `  W )
)
4931adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  F  e.  T )
505, 7, 8tendoidcl 31031 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
5150ad2antrr 706 . . . . . . . . . . . . . . . . 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 2285 . . . . . . . . . . . . . . . . . 18  |-  ( .s
`  U )  =  ( .s `  U
)
535, 7, 8, 41, 52dvhopvsca 31365 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . 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 4247 . . . . . . . . . . . . . . 15  |-  ( <.
g ,  s >.  =  <. ( x `  F ) ,  ( x  o.  (  _I  |`  T ) ) >.  <->  ( g  =  ( x `
 F )  /\  s  =  ( x  o.  (  _I  |`  T ) ) ) )
5755, 56syl6bb 252 . . . . . . . . . . . . . 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 31033 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W ) )  -> 
( x  o.  (  _I  |`  T ) )  =  x )
5958adantlr 695 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( ( TEndo `  K
) `  W )
)  ->  ( x  o.  (  _I  |`  T ) )  =  x )
6059eqeq2d 2296 . . . . . . . . . . . . . . . 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 1649 . . . . . . . . . . . . . . . 16  |-  ( s  =  x  <->  x  =  s )
6260, 61syl6bb 252 . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . 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 437 . . . . . . . . . . . . 13  |-  ( ( g  =  ( x `
 F )  /\  x  =  s )  <->  ( x  =  s  /\  g  =  ( x `  F ) ) )
6664, 65syl6bb 252 . . . . . . . . . . . 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 2562 . . . . . . . . . . 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 244 . . . . . . . . . 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 1258 . . . . . . . . 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 5526 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
x `  F )  =  ( s `  F ) )
7170eqeq2d 2296 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
g  =  ( x `
 F )  <->  g  =  ( s `  F
) ) )
7271ceqsrexv 2903 . . . . . . . . . . . 12  |-  ( s  e.  ( ( TEndo `  K ) `  W
)  ->  ( E. x  e.  ( ( TEndo `  K ) `  W ) ( x  =  s  /\  g  =  ( x `  F ) )  <->  g  =  ( s `  F
) ) )
7372pm5.32i 618 . . . . . . . . . . 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 675 . . . . . . . . . 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 938 . . . . . . . . . 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 938 . . . . . . . . . 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 268 . . . . . . . . 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 253 . . . . . . . 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 244 . . . . . . 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 2291 . . . . . . . . . . 11  |-  ( v  =  <. g ,  s
>.  ->  ( v  =  ( x ( .s
`  U ) <. F ,  (  _I  |`  T ) >. )  <->  <.
g ,  s >.  =  ( x ( .s `  U )
<. F ,  (  _I  |`  T ) >. )
) )
8180rexbidv 2566 . . . . . . . . . 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 4727 . . . . . . . . 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 2349 . . . . . . . 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 4273 . . . . . . . 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 241 . . . . . . 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 252 . . . . . 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 244 . . . . 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 4788 . . . 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 1181 . . 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 730 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9145eleq2d 2352 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( x  e.  (
Base `  (Scalar `  U
) )  <->  x  e.  ( ( TEndo `  K
) `  W )
) )
9291biimpa 470 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  x  e.  ( Base `  (Scalar `  U ) ) )  ->  x  e.  ( ( TEndo `  K ) `  W ) )
9350adantr 451 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
(  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
94 opelxpi 4723 . . . . . . . . . 10  |-  ( ( F  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )  ->  <. F , 
(  _I  |`  T )
>.  e.  ( T  X.  ( ( TEndo `  K
) `  W )
) )
9531, 93, 94syl2anc 642 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  <. F ,  (  _I  |`  T ) >.  e.  ( T  X.  ( (
TEndo `  K ) `  W ) ) )
9695adantr 451 . . . . . . . 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 31366 . . . . . . . 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 1180 . . . . . . 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 2354 . . . . . . 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 15 . . . . . 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 2669 . . . . 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 616 . . . 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 2399 . . 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 2343 . 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 31373 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  U  e.  LMod )
106 eqid 2285 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
1075, 7, 8, 41, 106dvhelvbasei 31351 . . . 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 1180 . . 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 15761 . . 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 642 . 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 2320 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 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   {cab 2271   E.wrex 2546   {crab 2549    C_ wss 3154   {csn 3642   <.cop 3645   class class class wbr 4025   {copab 4078    _I cid 4306    X. cxp 4689    |` cres 4693    o. ccom 4695   Rel wrel 4696   ` cfv 5257  (class class class)co 5860   iota_crio 6299   Basecbs 13150  Scalarcsca 13213   .scvsca 13214   lecple 13217   occoc 13218   LModclmod 15629   LSpanclspn 15730   Atomscatm 29526   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   TEndoctendo 31014   DVecHcdvh 31341   DIsoCcdic 31435
This theorem is referenced by:  cdlemn5pre  31463  dih1dimc  31505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tendo 31017  df-edring 31019  df-dvech 31342  df-dic 31436
  Copyright terms: Public domain W3C validator