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Theorem issh2 21781
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 21780 . 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 21573 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
3 ffun 5357 . . . . . . 7  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  Fun  +h  )
42, 3ax-mp 10 . . . . . 6  |-  Fun  +h
5 xpss12 4792 . . . . . . . 8  |-  ( ( H  C_  ~H  /\  H  C_ 
~H )  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
65anidms 628 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
72fdmi 5360 . . . . . . 7  |-  dom  +h  =  ( ~H  X.  ~H )
86, 7syl6sseqr 3227 . . . . . 6  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  dom  +h  )
9 funimassov 5959 . . . . . 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 646 . . . . 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 21578 . . . . . . 7  |-  .h  :
( CC  X.  ~H )
--> ~H
12 ffun 5357 . . . . . . 7  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  Fun  .h  )
1311, 12ax-mp 10 . . . . . 6  |-  Fun  .h
14 xpss2 4796 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  ( CC  X.  ~H )
)
1511fdmi 5360 . . . . . . 7  |-  dom  .h  =  ( CC  X.  ~H )
1614, 15syl6sseqr 3227 . . . . . 6  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  dom  .h  )
17 funimassov 5959 . . . . . 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 646 . . . . 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 693 . . . 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 453 . . 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 620 . 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 242 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 178    /\ wa 360    e. wcel 1685   A.wral 2545    C_ wss 3154    X. cxp 4687   dom cdm 4689   "cima 4692   Fun wfun 5216   -->wf 5218  (class class class)co 5820   CCcc 8731   ~Hchil 21492    +h cva 21493    .h csm 21494   0hc0v 21497   SHcsh 21501
This theorem is referenced by:  shaddcl  21789  shmulcl  21790  shmulclOLD  21791  issh3  21792  helch  21816  hsn0elch  21820  hhshsslem2  21838  ocsh  21855  shscli  21889  shintcli  21901  imaelshi  22631
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512  ax-hilex 21572  ax-hfvadd 21573  ax-hfvmul 21578
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-fv 5230  df-ov 5823  df-sh 21779
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