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Theorem sspval 21299
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  =  ( .s OLD `  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  /\  ( .s OLD `  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 5525 . . . . . . 7  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
3 sspval.g . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3syl6eqr 2333 . . . . . 6  |-  ( u  =  U  ->  ( +v `  u )  =  G )
54sseq2d 3206 . . . . 5  |-  ( u  =  U  ->  (
( +v `  w
)  C_  ( +v `  u )  <->  ( +v `  w )  C_  G
) )
6 fveq2 5525 . . . . . . 7  |-  ( u  =  U  ->  ( .s OLD `  u )  =  ( .s OLD `  U ) )
7 sspval.s . . . . . . 7  |-  S  =  ( .s OLD `  U
)
86, 7syl6eqr 2333 . . . . . 6  |-  ( u  =  U  ->  ( .s OLD `  u )  =  S )
98sseq2d 3206 . . . . 5  |-  ( u  =  U  ->  (
( .s OLD `  w
)  C_  ( .s OLD `  u )  <->  ( .s OLD `  w )  C_  S ) )
10 fveq2 5525 . . . . . . 7  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
11 sspval.n . . . . . . 7  |-  N  =  ( normCV `  U )
1210, 11syl6eqr 2333 . . . . . 6  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
1312sseq2d 3206 . . . . 5  |-  ( u  =  U  ->  (
( normCV `  w )  C_  ( normCV `  u )  <->  ( normCV `  w
)  C_  N )
)
145, 9, 133anbi123d 1252 . . . 4  |-  ( u  =  U  ->  (
( ( +v `  w )  C_  ( +v `  u )  /\  ( .s OLD `  w
)  C_  ( .s OLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) )  <-> 
( ( +v `  w )  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) ) )
1514rabbidv 2780 . . 3  |-  ( u  =  U  ->  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  ( +v `  u )  /\  ( .s OLD `  w ) 
C_  ( .s OLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) ) }  =  { w  e.  NrmCVec  |  ( ( +v
`  w )  C_  G  /\  ( .s OLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) } )
16 df-ssp 21298 . . 3  |-  SubSp  =  ( u  e.  NrmCVec  |->  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  ( +v `  u )  /\  ( .s OLD `  w ) 
C_  ( .s OLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) ) } )
17 fvex 5539 . . . . . . . 8  |-  ( +v
`  U )  e. 
_V
183, 17eqeltri 2353 . . . . . . 7  |-  G  e. 
_V
1918pwex 4193 . . . . . 6  |-  ~P G  e.  _V
20 fvex 5539 . . . . . . . 8  |-  ( .s
OLD `  U )  e.  _V
217, 20eqeltri 2353 . . . . . . 7  |-  S  e. 
_V
2221pwex 4193 . . . . . 6  |-  ~P S  e.  _V
2319, 22xpex 4801 . . . . 5  |-  ( ~P G  X.  ~P S
)  e.  _V
24 fvex 5539 . . . . . . 7  |-  ( normCV `  U )  e.  _V
2511, 24eqeltri 2353 . . . . . 6  |-  N  e. 
_V
2625pwex 4193 . . . . 5  |-  ~P N  e.  _V
2723, 26xpex 4801 . . . 4  |-  ( ( ~P G  X.  ~P S )  X.  ~P N )  e.  _V
28 rabss 3250 . . . . 5  |-  ( { w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) }  C_  ( ( ~P G  X.  ~P S )  X. 
~P N )  <->  A. w  e.  NrmCVec  ( ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
)  ->  w  e.  ( ( ~P G  X.  ~P S )  X. 
~P N ) ) )
29 fvex 5539 . . . . . . . . . 10  |-  ( +v
`  w )  e. 
_V
3029elpw 3631 . . . . . . . . 9  |-  ( ( +v `  w )  e.  ~P G  <->  ( +v `  w )  C_  G
)
31 fvex 5539 . . . . . . . . . 10  |-  ( .s
OLD `  w )  e.  _V
3231elpw 3631 . . . . . . . . 9  |-  ( ( .s OLD `  w
)  e.  ~P S  <->  ( .s OLD `  w
)  C_  S )
33 opelxpi 4721 . . . . . . . . 9  |-  ( ( ( +v `  w
)  e.  ~P G  /\  ( .s OLD `  w
)  e.  ~P S
)  ->  <. ( +v
`  w ) ,  ( .s OLD `  w
) >.  e.  ( ~P G  X.  ~P S
) )
3430, 32, 33syl2anbr 466 . . . . . . . 8  |-  ( ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S )  -> 
<. ( +v `  w
) ,  ( .s
OLD `  w ) >.  e.  ( ~P G  X.  ~P S ) )
35 fvex 5539 . . . . . . . . . 10  |-  ( normCV `  w )  e.  _V
3635elpw 3631 . . . . . . . . 9  |-  ( (
normCV
`  w )  e. 
~P N  <->  ( normCV `  w
)  C_  N )
3736biimpri 197 . . . . . . . 8  |-  ( (
normCV
`  w )  C_  N  ->  ( normCV `  w
)  e.  ~P N
)
38 opelxpi 4721 . . . . . . . 8  |-  ( (
<. ( +v `  w
) ,  ( .s
OLD `  w ) >.  e.  ( ~P G  X.  ~P S )  /\  ( normCV `  w )  e. 
~P N )  ->  <. <. ( +v `  w ) ,  ( .s OLD `  w
) >. ,  ( normCV `  w ) >.  e.  ( ( ~P G  X.  ~P S )  X.  ~P N ) )
3934, 37, 38syl2an 463 . . . . . . 7  |-  ( ( ( ( +v `  w )  C_  G  /\  ( .s OLD `  w
)  C_  S )  /\  ( normCV `  w )  C_  N )  ->  <. <. ( +v `  w ) ,  ( .s OLD `  w
) >. ,  ( normCV `  w ) >.  e.  ( ( ~P G  X.  ~P S )  X.  ~P N ) )
40393impa 1146 . . . . . 6  |-  ( ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N )  ->  <. <. ( +v `  w ) ,  ( .s OLD `  w
) >. ,  ( normCV `  w ) >.  e.  ( ( ~P G  X.  ~P S )  X.  ~P N ) )
41 eqid 2283 . . . . . . . 8  |-  ( +v
`  w )  =  ( +v `  w
)
42 eqid 2283 . . . . . . . 8  |-  ( .s
OLD `  w )  =  ( .s OLD `  w )
43 eqid 2283 . . . . . . . 8  |-  ( normCV `  w )  =  (
normCV
`  w )
4441, 42, 43nvop 21243 . . . . . . 7  |-  ( w  e.  NrmCVec  ->  w  =  <. <.
( +v `  w
) ,  ( .s
OLD `  w ) >. ,  ( normCV `  w
) >. )
4544eleq1d 2349 . . . . . 6  |-  ( w  e.  NrmCVec  ->  ( w  e.  ( ( ~P G  X.  ~P S )  X. 
~P N )  <->  <. <. ( +v `  w ) ,  ( .s OLD `  w
) >. ,  ( normCV `  w ) >.  e.  ( ( ~P G  X.  ~P S )  X.  ~P N ) ) )
4640, 45syl5ibr 212 . . . . 5  |-  ( w  e.  NrmCVec  ->  ( ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
)  ->  w  e.  ( ( ~P G  X.  ~P S )  X. 
~P N ) ) )
4728, 46mprgbir 2613 . . . 4  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  C_  (
( ~P G  X.  ~P S )  X.  ~P N )
4827, 47ssexi 4159 . . 3  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  e.  _V
4915, 16, 48fvmpt 5602 . 2  |-  ( U  e.  NrmCVec  ->  ( SubSp `  U
)  =  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } )
501, 49syl5eq 2327 1  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   <.cop 3643    X. cxp 4687   ` cfv 5255   NrmCVeccnv 21140   +vcpv 21141   .s OLDcns 21143   normCVcnmcv 21146   SubSpcss 21297
This theorem is referenced by:  isssp  21300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-sm 21153  df-nmcv 21156  df-ssp 21298
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