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

Theorem sshjval 22852
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 22502 . . 3  |-  ~H  e.  _V
21elpw2 4364 . 2  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4364 . 2  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uneq12 3496 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  u.  y
)  =  ( A  u.  B ) )
54fveq2d 5732 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  (
x  u.  y ) )  =  ( _|_ `  ( A  u.  B
) ) )
65fveq2d 5732 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
7 df-chj 22812 . . 3  |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
8 fvex 5742 . . 3  |-  ( _|_ `  ( _|_ `  ( A  u.  B )
) )  e.  _V
96, 7, 8ovmpt2a 6204 . 2  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
102, 3, 9syl2anbr 467 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3318    C_ wss 3320   ~Pcpw 3799   ` cfv 5454  (class class class)co 6081   ~Hchil 22422   _|_cort 22433    vH chj 22436
This theorem is referenced by:  shjval  22853  sshjval3  22856  sshjcl  22857  sshjval2  22913  ssjo  22949  sshhococi  23048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-chj 22812
  Copyright terms: Public domain W3C validator