HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  issh2 Unicode version

Theorem issh2 22664
Description: Subspace  H of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
issh2  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  ( A. x  e.  H  A. y  e.  H  ( x  +h  y
)  e.  H  /\  A. x  e.  CC  A. y  e.  H  (
x  .h  y )  e.  H ) ) )
Distinct variable group:    x, y, H

Proof of Theorem issh2
StepHypRef Expression
1 issh 22663 . 2  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
2 ax-hfvadd 22456 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
3 ffun 5552 . . . . . . 7  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  Fun  +h  )
42, 3ax-mp 8 . . . . . 6  |-  Fun  +h
5 xpss12 4940 . . . . . . . 8  |-  ( ( H  C_  ~H  /\  H  C_ 
~H )  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
65anidms 627 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
72fdmi 5555 . . . . . . 7  |-  dom  +h  =  ( ~H  X.  ~H )
86, 7syl6sseqr 3355 . . . . . 6  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  dom  +h  )
9 funimassov 6182 . . . . . 6  |-  ( ( Fun  +h  /\  ( H  X.  H )  C_  dom  +h  )  ->  (
(  +h  " ( H  X.  H ) ) 
C_  H  <->  A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H
) )
104, 8, 9sylancr 645 . . . . 5  |-  ( H 
C_  ~H  ->  ( (  +h  " ( H  X.  H ) ) 
C_  H  <->  A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H
) )
11 ax-hfvmul 22461 . . . . . . 7  |-  .h  :
( CC  X.  ~H )
--> ~H
12 ffun 5552 . . . . . . 7  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  Fun  .h  )
1311, 12ax-mp 8 . . . . . 6  |-  Fun  .h
14 xpss2 4944 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  ( CC  X.  ~H )
)
1511fdmi 5555 . . . . . . 7  |-  dom  .h  =  ( CC  X.  ~H )
1614, 15syl6sseqr 3355 . . . . . 6  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  dom  .h  )
17 funimassov 6182 . . . . . 6  |-  ( ( Fun  .h  /\  ( CC  X.  H )  C_  dom  .h  )  ->  (
(  .h  " ( CC  X.  H ) ) 
C_  H  <->  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
) )
1813, 16, 17sylancr 645 . . . . 5  |-  ( H 
C_  ~H  ->  ( (  .h  " ( CC 
X.  H ) ) 
C_  H  <->  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
) )
1910, 18anbi12d 692 . . . 4  |-  ( H 
C_  ~H  ->  ( ( (  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H )  <->  ( A. x  e.  H  A. y  e.  H  (
x  +h  y )  e.  H  /\  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
) ) )
2019adantr 452 . . 3  |-  ( ( H  C_  ~H  /\  0h  e.  H )  ->  (
( (  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H )  <->  ( A. x  e.  H  A. y  e.  H  (
x  +h  y )  e.  H  /\  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
) ) )
2120pm5.32i 619 . 2  |-  ( ( ( H  C_  ~H  /\ 
0h  e.  H )  /\  ( (  +h  " ( H  X.  H ) )  C_  H  /\  (  .h  "
( CC  X.  H
) )  C_  H
) )  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  ( A. x  e.  H  A. y  e.  H  ( x  +h  y
)  e.  H  /\  A. x  e.  CC  A. y  e.  H  (
x  .h  y )  e.  H ) ) )
221, 21bitri 241 1  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  ( A. x  e.  H  A. y  e.  H  ( x  +h  y
)  e.  H  /\  A. x  e.  CC  A. y  e.  H  (
x  .h  y )  e.  H ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   A.wral 2666    C_ wss 3280    X. cxp 4835   dom cdm 4837   "cima 4840   Fun wfun 5407   -->wf 5409  (class class class)co 6040   CCcc 8944   ~Hchil 22375    +h cva 22376    .h csm 22377   0hc0v 22380   SHcsh 22384
This theorem is referenced by:  shaddcl  22672  shmulcl  22673  shmulclOLD  22674  issh3  22675  helch  22699  hsn0elch  22703  hhshsslem2  22721  ocsh  22738  shscli  22772  shintcli  22784  imaelshi  23514
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-hilex 22455  ax-hfvadd 22456  ax-hfvmul 22461
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-sh 22662
  Copyright terms: Public domain W3C validator