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Theorem sspval 24121
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 5691 . . . . . . 7  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
3 sspval.g . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3syl6eqr 2493 . . . . . 6  |-  ( u  =  U  ->  ( +v `  u )  =  G )
54sseq2d 3384 . . . . 5  |-  ( u  =  U  ->  (
( +v `  w
)  C_  ( +v `  u )  <->  ( +v `  w )  C_  G
) )
6 fveq2 5691 . . . . . . 7  |-  ( u  =  U  ->  ( .sOLD `  u )  =  ( .sOLD `  U ) )
7 sspval.s . . . . . . 7  |-  S  =  ( .sOLD `  U )
86, 7syl6eqr 2493 . . . . . 6  |-  ( u  =  U  ->  ( .sOLD `  u )  =  S )
98sseq2d 3384 . . . . 5  |-  ( u  =  U  ->  (
( .sOLD `  w )  C_  ( .sOLD `  u )  <-> 
( .sOLD `  w )  C_  S
) )
10 fveq2 5691 . . . . . . 7  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
11 sspval.n . . . . . . 7  |-  N  =  ( normCV `  U )
1210, 11syl6eqr 2493 . . . . . 6  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
1312sseq2d 3384 . . . . 5  |-  ( u  =  U  ->  (
( normCV `  w )  C_  ( normCV `  u )  <->  ( normCV `  w
)  C_  N )
)
145, 9, 133anbi123d 1289 . . . 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 2964 . . 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 24120 . . 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 5701 . . . . . . . 8  |-  ( +v
`  U )  e. 
_V
183, 17eqeltri 2513 . . . . . . 7  |-  G  e. 
_V
1918pwex 4475 . . . . . 6  |-  ~P G  e.  _V
20 fvex 5701 . . . . . . . 8  |-  ( .sOLD `  U )  e.  _V
217, 20eqeltri 2513 . . . . . . 7  |-  S  e. 
_V
2221pwex 4475 . . . . . 6  |-  ~P S  e.  _V
2319, 22xpex 6508 . . . . 5  |-  ( ~P G  X.  ~P S
)  e.  _V
24 fvex 5701 . . . . . . 7  |-  ( normCV `  U )  e.  _V
2511, 24eqeltri 2513 . . . . . 6  |-  N  e. 
_V
2625pwex 4475 . . . . 5  |-  ~P N  e.  _V
2723, 26xpex 6508 . . . 4  |-  ( ( ~P G  X.  ~P S )  X.  ~P N )  e.  _V
28 rabss 3429 . . . . 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 5701 . . . . . . . . . 10  |-  ( +v
`  w )  e. 
_V
3029elpw 3866 . . . . . . . . 9  |-  ( ( +v `  w )  e.  ~P G  <->  ( +v `  w )  C_  G
)
31 fvex 5701 . . . . . . . . . 10  |-  ( .sOLD `  w )  e.  _V
3231elpw 3866 . . . . . . . . 9  |-  ( ( .sOLD `  w
)  e.  ~P S  <->  ( .sOLD `  w
)  C_  S )
33 opelxpi 4871 . . . . . . . . 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 5701 . . . . . . . . . 10  |-  ( normCV `  w )  e.  _V
3635elpw 3866 . . . . . . . . 9  |-  ( (
normCV
`  w )  e. 
~P N  <->  ( normCV `  w
)  C_  N )
3736biimpri 206 . . . . . . . 8  |-  ( (
normCV
`  w )  C_  N  ->  ( normCV `  w
)  e.  ~P N
)
38 opelxpi 4871 . . . . . . . 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 1182 . . . . . 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 2443 . . . . . . . 8  |-  ( +v
`  w )  =  ( +v `  w
)
42 eqid 2443 . . . . . . . 8  |-  ( .sOLD `  w )  =  ( .sOLD `  w )
43 eqid 2443 . . . . . . . 8  |-  ( normCV `  w )  =  (
normCV
`  w )
4441, 42, 43nvop 24065 . . . . . . 7  |-  ( w  e.  NrmCVec  ->  w  =  <. <.
( +v `  w
) ,  ( .sOLD `  w )
>. ,  ( normCV `  w
) >. )
4544eleq1d 2509 . . . . . 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 2786 . . . 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 4437 . . 3  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  e.  _V
4915, 16, 48fvmpt 5774 . 2  |-  ( U  e.  NrmCVec  ->  ( SubSp `  U
)  =  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } )
501, 49syl5eq 2487 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 965    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   <.cop 3883    X. cxp 4838   ` cfv 5418   NrmCVeccnv 23962   +vcpv 23963   .sOLDcns 23965   normCVcnmcv 23968   SubSpcss 24119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-oprab 6095  df-1st 6577  df-2nd 6578  df-vc 23924  df-nv 23970  df-va 23973  df-sm 23975  df-nmcv 23978  df-ssp 24120
This theorem is referenced by:  isssp  24122
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