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Theorem sspval 25467
Description: The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspval.g  |-  G  =  ( +v `  U
)
sspval.s  |-  S  =  ( .sOLD `  U )
sspval.n  |-  N  =  ( normCV `  U )
sspval.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspval  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
Distinct variable groups:    w, G    w, N    w, S    w, U
Allowed substitution hint:    H( w)

Proof of Theorem sspval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 sspval.h . 2  |-  H  =  ( SubSp `  U )
2 fveq2 5872 . . . . . . 7  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
3 sspval.g . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3syl6eqr 2526 . . . . . 6  |-  ( u  =  U  ->  ( +v `  u )  =  G )
54sseq2d 3537 . . . . 5  |-  ( u  =  U  ->  (
( +v `  w
)  C_  ( +v `  u )  <->  ( +v `  w )  C_  G
) )
6 fveq2 5872 . . . . . . 7  |-  ( u  =  U  ->  ( .sOLD `  u )  =  ( .sOLD `  U ) )
7 sspval.s . . . . . . 7  |-  S  =  ( .sOLD `  U )
86, 7syl6eqr 2526 . . . . . 6  |-  ( u  =  U  ->  ( .sOLD `  u )  =  S )
98sseq2d 3537 . . . . 5  |-  ( u  =  U  ->  (
( .sOLD `  w )  C_  ( .sOLD `  u )  <-> 
( .sOLD `  w )  C_  S
) )
10 fveq2 5872 . . . . . . 7  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
11 sspval.n . . . . . . 7  |-  N  =  ( normCV `  U )
1210, 11syl6eqr 2526 . . . . . 6  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
1312sseq2d 3537 . . . . 5  |-  ( u  =  U  ->  (
( normCV `  w )  C_  ( normCV `  u )  <->  ( normCV `  w
)  C_  N )
)
145, 9, 133anbi123d 1299 . . . 4  |-  ( u  =  U  ->  (
( ( +v `  w )  C_  ( +v `  u )  /\  ( .sOLD `  w
)  C_  ( .sOLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) )  <-> 
( ( +v `  w )  C_  G  /\  ( .sOLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) ) )
1514rabbidv 3110 . . 3  |-  ( u  =  U  ->  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  ( +v `  u )  /\  ( .sOLD `  w ) 
C_  ( .sOLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) ) }  =  { w  e.  NrmCVec  |  ( ( +v
`  w )  C_  G  /\  ( .sOLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) } )
16 df-ssp 25466 . . 3  |-  SubSp  =  ( u  e.  NrmCVec  |->  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  ( +v `  u )  /\  ( .sOLD `  w ) 
C_  ( .sOLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) ) } )
17 fvex 5882 . . . . . . . 8  |-  ( +v
`  U )  e. 
_V
183, 17eqeltri 2551 . . . . . . 7  |-  G  e. 
_V
1918pwex 4636 . . . . . 6  |-  ~P G  e.  _V
20 fvex 5882 . . . . . . . 8  |-  ( .sOLD `  U )  e.  _V
217, 20eqeltri 2551 . . . . . . 7  |-  S  e. 
_V
2221pwex 4636 . . . . . 6  |-  ~P S  e.  _V
2319, 22xpex 6599 . . . . 5  |-  ( ~P G  X.  ~P S
)  e.  _V
24 fvex 5882 . . . . . . 7  |-  ( normCV `  U )  e.  _V
2511, 24eqeltri 2551 . . . . . 6  |-  N  e. 
_V
2625pwex 4636 . . . . 5  |-  ~P N  e.  _V
2723, 26xpex 6599 . . . 4  |-  ( ( ~P G  X.  ~P S )  X.  ~P N )  e.  _V
28 rabss 3582 . . . . 5  |-  ( { w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) }  C_  ( ( ~P G  X.  ~P S )  X. 
~P N )  <->  A. w  e.  NrmCVec  ( ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
)  ->  w  e.  ( ( ~P G  X.  ~P S )  X. 
~P N ) ) )
29 fvex 5882 . . . . . . . . . 10  |-  ( +v
`  w )  e. 
_V
3029elpw 4022 . . . . . . . . 9  |-  ( ( +v `  w )  e.  ~P G  <->  ( +v `  w )  C_  G
)
31 fvex 5882 . . . . . . . . . 10  |-  ( .sOLD `  w )  e.  _V
3231elpw 4022 . . . . . . . . 9  |-  ( ( .sOLD `  w
)  e.  ~P S  <->  ( .sOLD `  w
)  C_  S )
33 opelxpi 5037 . . . . . . . . 9  |-  ( ( ( +v `  w
)  e.  ~P G  /\  ( .sOLD `  w )  e.  ~P S )  ->  <. ( +v `  w ) ,  ( .sOLD `  w ) >.  e.  ( ~P G  X.  ~P S ) )
3430, 32, 33syl2anbr 480 . . . . . . . 8  |-  ( ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S )  -> 
<. ( +v `  w
) ,  ( .sOLD `  w )
>.  e.  ( ~P G  X.  ~P S ) )
35 fvex 5882 . . . . . . . . . 10  |-  ( normCV `  w )  e.  _V
3635elpw 4022 . . . . . . . . 9  |-  ( (
normCV
`  w )  e. 
~P N  <->  ( normCV `  w
)  C_  N )
3736biimpri 206 . . . . . . . 8  |-  ( (
normCV
`  w )  C_  N  ->  ( normCV `  w
)  e.  ~P N
)
38 opelxpi 5037 . . . . . . . 8  |-  ( (
<. ( +v `  w
) ,  ( .sOLD `  w )
>.  e.  ( ~P G  X.  ~P S )  /\  ( normCV `  w )  e. 
~P N )  ->  <. <. ( +v `  w ) ,  ( .sOLD `  w
) >. ,  ( normCV `  w ) >.  e.  ( ( ~P G  X.  ~P S )  X.  ~P N ) )
3934, 37, 38syl2an 477 . . . . . . 7  |-  ( ( ( ( +v `  w )  C_  G  /\  ( .sOLD `  w )  C_  S
)  /\  ( normCV `  w
)  C_  N )  -> 
<. <. ( +v `  w ) ,  ( .sOLD `  w
) >. ,  ( normCV `  w ) >.  e.  ( ( ~P G  X.  ~P S )  X.  ~P N ) )
40393impa 1191 . . . . . 6  |-  ( ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N )  ->  <. <. ( +v `  w ) ,  ( .sOLD `  w ) >. ,  (
normCV
`  w ) >.  e.  ( ( ~P G  X.  ~P S )  X. 
~P N ) )
41 eqid 2467 . . . . . . . 8  |-  ( +v
`  w )  =  ( +v `  w
)
42 eqid 2467 . . . . . . . 8  |-  ( .sOLD `  w )  =  ( .sOLD `  w )
43 eqid 2467 . . . . . . . 8  |-  ( normCV `  w )  =  (
normCV
`  w )
4441, 42, 43nvop 25411 . . . . . . 7  |-  ( w  e.  NrmCVec  ->  w  =  <. <.
( +v `  w
) ,  ( .sOLD `  w )
>. ,  ( normCV `  w
) >. )
4544eleq1d 2536 . . . . . 6  |-  ( w  e.  NrmCVec  ->  ( w  e.  ( ( ~P G  X.  ~P S )  X. 
~P N )  <->  <. <. ( +v `  w ) ,  ( .sOLD `  w ) >. ,  (
normCV
`  w ) >.  e.  ( ( ~P G  X.  ~P S )  X. 
~P N ) ) )
4640, 45syl5ibr 221 . . . . 5  |-  ( w  e.  NrmCVec  ->  ( ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
)  ->  w  e.  ( ( ~P G  X.  ~P S )  X. 
~P N ) ) )
4728, 46mprgbir 2831 . . . 4  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  C_  (
( ~P G  X.  ~P S )  X.  ~P N )
4827, 47ssexi 4598 . . 3  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  e.  _V
4915, 16, 48fvmpt 5957 . 2  |-  ( U  e.  NrmCVec  ->  ( SubSp `  U
)  =  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } )
501, 49syl5eq 2520 1  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   <.cop 4039    X. cxp 5003   ` cfv 5594   NrmCVeccnv 25308   +vcpv 25309   .sOLDcns 25311   normCVcnmcv 25314   SubSpcss 25465
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-oprab 6299  df-1st 6795  df-2nd 6796  df-vc 25270  df-nv 25316  df-va 25319  df-sm 25321  df-nmcv 25324  df-ssp 25466
This theorem is referenced by:  isssp  25468
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