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Theorem sspval 22214
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 5720 . . . . . . 7  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
3 sspval.g . . . . . . 7  |-  G  =  ( +v `  U
)
42, 3syl6eqr 2485 . . . . . 6  |-  ( u  =  U  ->  ( +v `  u )  =  G )
54sseq2d 3368 . . . . 5  |-  ( u  =  U  ->  (
( +v `  w
)  C_  ( +v `  u )  <->  ( +v `  w )  C_  G
) )
6 fveq2 5720 . . . . . . 7  |-  ( u  =  U  ->  ( .s OLD `  u )  =  ( .s OLD `  U ) )
7 sspval.s . . . . . . 7  |-  S  =  ( .s OLD `  U
)
86, 7syl6eqr 2485 . . . . . 6  |-  ( u  =  U  ->  ( .s OLD `  u )  =  S )
98sseq2d 3368 . . . . 5  |-  ( u  =  U  ->  (
( .s OLD `  w
)  C_  ( .s OLD `  u )  <->  ( .s OLD `  w )  C_  S ) )
10 fveq2 5720 . . . . . . 7  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
11 sspval.n . . . . . . 7  |-  N  =  ( normCV `  U )
1210, 11syl6eqr 2485 . . . . . 6  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
1312sseq2d 3368 . . . . 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 2940 . . 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 22213 . . 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 5734 . . . . . . . 8  |-  ( +v
`  U )  e. 
_V
183, 17eqeltri 2505 . . . . . . 7  |-  G  e. 
_V
1918pwex 4374 . . . . . 6  |-  ~P G  e.  _V
20 fvex 5734 . . . . . . . 8  |-  ( .s
OLD `  U )  e.  _V
217, 20eqeltri 2505 . . . . . . 7  |-  S  e. 
_V
2221pwex 4374 . . . . . 6  |-  ~P S  e.  _V
2319, 22xpex 4982 . . . . 5  |-  ( ~P G  X.  ~P S
)  e.  _V
24 fvex 5734 . . . . . . 7  |-  ( normCV `  U )  e.  _V
2511, 24eqeltri 2505 . . . . . 6  |-  N  e. 
_V
2625pwex 4374 . . . . 5  |-  ~P N  e.  _V
2723, 26xpex 4982 . . . 4  |-  ( ( ~P G  X.  ~P S )  X.  ~P N )  e.  _V
28 rabss 3412 . . . . 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 5734 . . . . . . . . . 10  |-  ( +v
`  w )  e. 
_V
3029elpw 3797 . . . . . . . . 9  |-  ( ( +v `  w )  e.  ~P G  <->  ( +v `  w )  C_  G
)
31 fvex 5734 . . . . . . . . . 10  |-  ( .s
OLD `  w )  e.  _V
3231elpw 3797 . . . . . . . . 9  |-  ( ( .s OLD `  w
)  e.  ~P S  <->  ( .s OLD `  w
)  C_  S )
33 opelxpi 4902 . . . . . . . . 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 5734 . . . . . . . . . 10  |-  ( normCV `  w )  e.  _V
3635elpw 3797 . . . . . . . . 9  |-  ( (
normCV
`  w )  e. 
~P N  <->  ( normCV `  w
)  C_  N )
3736biimpri 198 . . . . . . . 8  |-  ( (
normCV
`  w )  C_  N  ->  ( normCV `  w
)  e.  ~P N
)
38 opelxpi 4902 . . . . . . . 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 2435 . . . . . . . 8  |-  ( +v
`  w )  =  ( +v `  w
)
42 eqid 2435 . . . . . . . 8  |-  ( .s
OLD `  w )  =  ( .s OLD `  w )
43 eqid 2435 . . . . . . . 8  |-  ( normCV `  w )  =  (
normCV
`  w )
4441, 42, 43nvop 22158 . . . . . . 7  |-  ( w  e.  NrmCVec  ->  w  =  <. <.
( +v `  w
) ,  ( .s
OLD `  w ) >. ,  ( normCV `  w
) >. )
4544eleq1d 2501 . . . . . 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 2768 . . . 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 4340 . . 3  |-  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) }  e.  _V
4915, 16, 48fvmpt 5798 . 2  |-  ( U  e.  NrmCVec  ->  ( SubSp `  U
)  =  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } )
501, 49syl5eq 2479 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   <.cop 3809    X. cxp 4868   ` cfv 5446   NrmCVeccnv 22055   +vcpv 22056   .s OLDcns 22058   normCVcnmcv 22061   SubSpcss 22212
This theorem is referenced by:  isssp  22215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-sm 22068  df-nmcv 22071  df-ssp 22213
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