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Theorem issh2 22560
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 22559 . 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 22352 . . . . . . 7  |-  +h  :
( ~H  X.  ~H )
--> ~H
3 ffun 5534 . . . . . . 7  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  Fun  +h  )
42, 3ax-mp 8 . . . . . 6  |-  Fun  +h
5 xpss12 4922 . . . . . . . 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 5537 . . . . . . 7  |-  dom  +h  =  ( ~H  X.  ~H )
86, 7syl6sseqr 3339 . . . . . 6  |-  ( H 
C_  ~H  ->  ( H  X.  H )  C_  dom  +h  )
9 funimassov 6163 . . . . . 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 22357 . . . . . . 7  |-  .h  :
( CC  X.  ~H )
--> ~H
12 ffun 5534 . . . . . . 7  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  Fun  .h  )
1311, 12ax-mp 8 . . . . . 6  |-  Fun  .h
14 xpss2 4926 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  ( CC  X.  ~H )
)
1511fdmi 5537 . . . . . . 7  |-  dom  .h  =  ( CC  X.  ~H )
1614, 15syl6sseqr 3339 . . . . . 6  |-  ( H 
C_  ~H  ->  ( CC 
X.  H )  C_  dom  .h  )
17 funimassov 6163 . . . . . 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 1717   A.wral 2650    C_ wss 3264    X. cxp 4817   dom cdm 4819   "cima 4822   Fun wfun 5389   -->wf 5391  (class class class)co 6021   CCcc 8922   ~Hchil 22271    +h cva 22272    .h csm 22273   0hc0v 22276   SHcsh 22280
This theorem is referenced by:  shaddcl  22568  shmulcl  22569  shmulclOLD  22570  issh3  22571  helch  22595  hsn0elch  22599  hhshsslem2  22617  ocsh  22634  shscli  22668  shintcli  22680  imaelshi  23410
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-hilex 22351  ax-hfvadd 22352  ax-hfvmul 22357
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-csb 3196  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-iun 4038  df-br 4155  df-opab 4209  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-fv 5403  df-ov 6024  df-sh 22558
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