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Theorem sshjval 26988
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 26637 . . 3  |-  ~H  e.  _V
21elpw2 4584 . 2  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4584 . 2  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uneq12 3615 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  u.  y
)  =  ( A  u.  B ) )
54fveq2d 5881 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  (
x  u.  y ) )  =  ( _|_ `  ( A  u.  B
) ) )
65fveq2d 5881 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
7 df-chj 26948 . . 3  |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
8 fvex 5887 . . 3  |-  ( _|_ `  ( _|_ `  ( A  u.  B )
) )  e.  _V
96, 7, 8ovmpt2a 6437 . 2  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
102, 3, 9syl2anbr 482 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    u. cun 3434    C_ wss 3436   ~Pcpw 3979   ` cfv 5597  (class class class)co 6301   ~Hchil 26557   _|_cort 26568    vH chj 26571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-chj 26948
This theorem is referenced by:  shjval  26989  sshjval3  26992  sshjcl  26993  sshjval2  27049  ssjo  27085  sshhococi  27184
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