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

Theorem sshjval 26385
Description: Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
sshjval  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )

Proof of Theorem sshjval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 26033 . . 3  |-  ~H  e.  _V
21elpw2 4529 . 2  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4529 . 2  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uneq12 3567 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  u.  y
)  =  ( A  u.  B ) )
54fveq2d 5778 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  (
x  u.  y ) )  =  ( _|_ `  ( A  u.  B
) ) )
65fveq2d 5778 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
7 df-chj 26345 . . 3  |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
8 fvex 5784 . . 3  |-  ( _|_ `  ( _|_ `  ( A  u.  B )
) )  e.  _V
96, 7, 8ovmpt2a 6332 . 2  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
102, 3, 9syl2anbr 478 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    u. cun 3387    C_ wss 3389   ~Pcpw 3927   ` cfv 5496  (class class class)co 6196   ~Hchil 25953   _|_cort 25964    vH chj 25967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-hilex 26033
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-chj 26345
This theorem is referenced by:  shjval  26386  sshjval3  26389  sshjcl  26390  sshjval2  26446  ssjo  26482  sshhococi  26581
  Copyright terms: Public domain W3C validator