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Theorem sspval 22071
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 5669 . . . . . . 7  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
3 sspval.g . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3syl6eqr 2438 . . . . . 6  |-  ( u  =  U  ->  ( +v `  u )  =  G )
54sseq2d 3320 . . . . 5  |-  ( u  =  U  ->  (
( +v `  w
)  C_  ( +v `  u )  <->  ( +v `  w )  C_  G
) )
6 fveq2 5669 . . . . . . 7  |-  ( u  =  U  ->  ( .s OLD `  u )  =  ( .s OLD `  U ) )
7 sspval.s . . . . . . 7  |-  S  =  ( .s OLD `  U
)
86, 7syl6eqr 2438 . . . . . 6  |-  ( u  =  U  ->  ( .s OLD `  u )  =  S )
98sseq2d 3320 . . . . 5  |-  ( u  =  U  ->  (
( .s OLD `  w
)  C_  ( .s OLD `  u )  <->  ( .s OLD `  w )  C_  S ) )
10 fveq2 5669 . . . . . . 7  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
11 sspval.n . . . . . . 7  |-  N  =  ( normCV `  U )
1210, 11syl6eqr 2438 . . . . . 6  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
1312sseq2d 3320 . . . . 5  |-  ( u  =  U  ->  (
( normCV `  w )  C_  ( normCV `  u )  <->  ( normCV `  w
)  C_  N )
)
145, 9, 133anbi123d 1254 . . . 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 2892 . . 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 22070 . . 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 5683 . . . . . . . 8  |-  ( +v
`  U )  e. 
_V
183, 17eqeltri 2458 . . . . . . 7  |-  G  e. 
_V
1918pwex 4324 . . . . . 6  |-  ~P G  e.  _V
20 fvex 5683 . . . . . . . 8  |-  ( .s
OLD `  U )  e.  _V
217, 20eqeltri 2458 . . . . . . 7  |-  S  e. 
_V
2221pwex 4324 . . . . . 6  |-  ~P S  e.  _V
2319, 22xpex 4931 . . . . 5  |-  ( ~P G  X.  ~P S
)  e.  _V
24 fvex 5683 . . . . . . 7  |-  ( normCV `  U )  e.  _V
2511, 24eqeltri 2458 . . . . . 6  |-  N  e. 
_V
2625pwex 4324 . . . . 5  |-  ~P N  e.  _V
2723, 26xpex 4931 . . . 4  |-  ( ( ~P G  X.  ~P S )  X.  ~P N )  e.  _V
28 rabss 3364 . . . . 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 5683 . . . . . . . . . 10  |-  ( +v
`  w )  e. 
_V
3029elpw 3749 . . . . . . . . 9  |-  ( ( +v `  w )  e.  ~P G  <->  ( +v `  w )  C_  G
)
31 fvex 5683 . . . . . . . . . 10  |-  ( .s
OLD `  w )  e.  _V
3231elpw 3749 . . . . . . . . 9  |-  ( ( .s OLD `  w
)  e.  ~P S  <->  ( .s OLD `  w
)  C_  S )
33 opelxpi 4851 . . . . . . . . 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 467 . . . . . . . 8  |-  ( ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S )  -> 
<. ( +v `  w
) ,  ( .s
OLD `  w ) >.  e.  ( ~P G  X.  ~P S ) )
35 fvex 5683 . . . . . . . . . 10  |-  ( normCV `  w )  e.  _V
3635elpw 3749 . . . . . . . . 9  |-  ( (
normCV
`  w )  e. 
~P N  <->  ( normCV `  w
)  C_  N )
3736biimpri 198 . . . . . . . 8  |-  ( (
normCV
`  w )  C_  N  ->  ( normCV `  w
)  e.  ~P N
)
38 opelxpi 4851 . . . . . . . 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 464 . . . . . . 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 1148 . . . . . 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 2388 . . . . . . . 8  |-  ( +v
`  w )  =  ( +v `  w
)
42 eqid 2388 . . . . . . . 8  |-  ( .s
OLD `  w )  =  ( .s OLD `  w )
43 eqid 2388 . . . . . . . 8  |-  ( normCV `  w )  =  (
normCV
`  w )
4441, 42, 43nvop 22015 . . . . . . 7  |-  ( w  e.  NrmCVec  ->  w  =  <. <.
( +v `  w
) ,  ( .s
OLD `  w ) >. ,  ( normCV `  w
) >. )
4544eleq1d 2454 . . . . . 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 213 . . . . 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 2720 . . . 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 4290 . . 3  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  e.  _V
4915, 16, 48fvmpt 5746 . 2  |-  ( U  e.  NrmCVec  ->  ( SubSp `  U
)  =  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } )
501, 49syl5eq 2432 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900    C_ wss 3264   ~Pcpw 3743   <.cop 3761    X. cxp 4817   ` cfv 5395   NrmCVeccnv 21912   +vcpv 21913   .s OLDcns 21915   normCVcnmcv 21918   SubSpcss 22069
This theorem is referenced by:  isssp  22072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-oprab 6025  df-1st 6289  df-2nd 6290  df-vc 21874  df-nv 21920  df-va 21923  df-sm 21925  df-nmcv 21928  df-ssp 22070
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