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Theorem sspval 26345
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 5877 . . . . . . 7  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
3 sspval.g . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3syl6eqr 2481 . . . . . 6  |-  ( u  =  U  ->  ( +v `  u )  =  G )
54sseq2d 3492 . . . . 5  |-  ( u  =  U  ->  (
( +v `  w
)  C_  ( +v `  u )  <->  ( +v `  w )  C_  G
) )
6 fveq2 5877 . . . . . . 7  |-  ( u  =  U  ->  ( .sOLD `  u )  =  ( .sOLD `  U ) )
7 sspval.s . . . . . . 7  |-  S  =  ( .sOLD `  U )
86, 7syl6eqr 2481 . . . . . 6  |-  ( u  =  U  ->  ( .sOLD `  u )  =  S )
98sseq2d 3492 . . . . 5  |-  ( u  =  U  ->  (
( .sOLD `  w )  C_  ( .sOLD `  u )  <-> 
( .sOLD `  w )  C_  S
) )
10 fveq2 5877 . . . . . . 7  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
11 sspval.n . . . . . . 7  |-  N  =  ( normCV `  U )
1210, 11syl6eqr 2481 . . . . . 6  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
1312sseq2d 3492 . . . . 5  |-  ( u  =  U  ->  (
( normCV `  w )  C_  ( normCV `  u )  <->  ( normCV `  w
)  C_  N )
)
145, 9, 133anbi123d 1335 . . . 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 3072 . . 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 26344 . . 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 5887 . . . . . . . 8  |-  ( +v
`  U )  e. 
_V
183, 17eqeltri 2506 . . . . . . 7  |-  G  e. 
_V
1918pwex 4603 . . . . . 6  |-  ~P G  e.  _V
20 fvex 5887 . . . . . . . 8  |-  ( .sOLD `  U )  e.  _V
217, 20eqeltri 2506 . . . . . . 7  |-  S  e. 
_V
2221pwex 4603 . . . . . 6  |-  ~P S  e.  _V
2319, 22xpex 6605 . . . . 5  |-  ( ~P G  X.  ~P S
)  e.  _V
24 fvex 5887 . . . . . . 7  |-  ( normCV `  U )  e.  _V
2511, 24eqeltri 2506 . . . . . 6  |-  N  e. 
_V
2625pwex 4603 . . . . 5  |-  ~P N  e.  _V
2723, 26xpex 6605 . . . 4  |-  ( ( ~P G  X.  ~P S )  X.  ~P N )  e.  _V
28 rabss 3538 . . . . 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 5887 . . . . . . . . . 10  |-  ( +v
`  w )  e. 
_V
3029elpw 3985 . . . . . . . . 9  |-  ( ( +v `  w )  e.  ~P G  <->  ( +v `  w )  C_  G
)
31 fvex 5887 . . . . . . . . . 10  |-  ( .sOLD `  w )  e.  _V
3231elpw 3985 . . . . . . . . 9  |-  ( ( .sOLD `  w
)  e.  ~P S  <->  ( .sOLD `  w
)  C_  S )
33 opelxpi 4881 . . . . . . . . 9  |-  ( ( ( +v `  w
)  e.  ~P G  /\  ( .sOLD `  w )  e.  ~P S )  ->  <. ( +v `  w ) ,  ( .sOLD `  w ) >.  e.  ( ~P G  X.  ~P S ) )
3430, 32, 33syl2anbr 482 . . . . . . . 8  |-  ( ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S )  -> 
<. ( +v `  w
) ,  ( .sOLD `  w )
>.  e.  ( ~P G  X.  ~P S ) )
35 fvex 5887 . . . . . . . . . 10  |-  ( normCV `  w )  e.  _V
3635elpw 3985 . . . . . . . . 9  |-  ( (
normCV
`  w )  e. 
~P N  <->  ( normCV `  w
)  C_  N )
3736biimpri 209 . . . . . . . 8  |-  ( (
normCV
`  w )  C_  N  ->  ( normCV `  w
)  e.  ~P N
)
38 opelxpi 4881 . . . . . . . 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 479 . . . . . . 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 1200 . . . . . 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 2422 . . . . . . . 8  |-  ( +v
`  w )  =  ( +v `  w
)
42 eqid 2422 . . . . . . . 8  |-  ( .sOLD `  w )  =  ( .sOLD `  w )
43 eqid 2422 . . . . . . . 8  |-  ( normCV `  w )  =  (
normCV
`  w )
4441, 42, 43nvop 26289 . . . . . . 7  |-  ( w  e.  NrmCVec  ->  w  =  <. <.
( +v `  w
) ,  ( .sOLD `  w )
>. ,  ( normCV `  w
) >. )
4544eleq1d 2491 . . . . . 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 224 . . . . 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 2789 . . . 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 4565 . . 3  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  e.  _V
4915, 16, 48fvmpt 5960 . 2  |-  ( U  e.  NrmCVec  ->  ( SubSp `  U
)  =  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } )
501, 49syl5eq 2475 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868   {crab 2779   _Vcvv 3081    C_ wss 3436   ~Pcpw 3979   <.cop 4002    X. cxp 4847   ` cfv 5597   NrmCVeccnv 26186   +vcpv 26187   .sOLDcns 26189   normCVcnmcv 26192   SubSpcss 26343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fo 5603  df-fv 5605  df-oprab 6305  df-1st 6803  df-2nd 6804  df-vc 26148  df-nv 26194  df-va 26197  df-sm 26199  df-nmcv 26202  df-ssp 26344
This theorem is referenced by:  isssp  26346
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