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

Theorem issh2 26324
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 26323 . 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 26115 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
3 ffun 5715 . . . . . . 7  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  Fun  +h  )
42, 3ax-mp 5 . . . . . 6  |-  Fun  +h
5 xpss12 5096 . . . . . . . 8  |-  ( ( H  C_  ~H  /\  H  C_ 
~H )  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
65anidms 643 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
72fdmi 5718 . . . . . . 7  |-  dom  +h  =  ( ~H  X.  ~H )
86, 7syl6sseqr 3536 . . . . . 6  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  dom  +h  )
9 funimassov 6425 . . . . . 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 661 . . . . 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 26120 . . . . . . 7  |-  .h  :
( CC  X.  ~H )
--> ~H
12 ffun 5715 . . . . . . 7  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  Fun  .h  )
1311, 12ax-mp 5 . . . . . 6  |-  Fun  .h
14 xpss2 5100 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  ( CC  X.  ~H )
)
1511fdmi 5718 . . . . . . 7  |-  dom  .h  =  ( CC  X.  ~H )
1614, 15syl6sseqr 3536 . . . . . 6  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  dom  .h  )
17 funimassov 6425 . . . . . 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 661 . . . . 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 708 . . . 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 463 . . 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 635 . 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 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1823   A.wral 2804    C_ wss 3461    X. cxp 4986   dom cdm 4988   "cima 4991   Fun wfun 5564   -->wf 5566  (class class class)co 6270   CCcc 9479   ~Hchil 26034    +h cva 26035    .h csm 26036   0hc0v 26039   SHcsh 26043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-hilex 26114  ax-hfvadd 26115  ax-hfvmul 26120
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-sh 26322
This theorem is referenced by:  shaddcl  26332  shmulcl  26333  shmulclOLD  26334  issh3  26335  helch  26359  hsn0elch  26364  hhshsslem2  26382  ocsh  26399  shscli  26433  shintcli  26445  imaelshi  27175
  Copyright terms: Public domain W3C validator