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Theorem sshjval 24900
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 24548 . . 3  |-  ~H  e.  _V
21elpw2 4559 . 2  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4559 . 2  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uneq12 3608 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  u.  y
)  =  ( A  u.  B ) )
54fveq2d 5798 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  (
x  u.  y ) )  =  ( _|_ `  ( A  u.  B
) ) )
65fveq2d 5798 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
7 df-chj 24860 . . 3  |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
8 fvex 5804 . . 3  |-  ( _|_ `  ( _|_ `  ( A  u.  B )
) )  e.  _V
96, 7, 8ovmpt2a 6326 . 2  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
102, 3, 9syl2anbr 480 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3429    C_ wss 3431   ~Pcpw 3963   ` cfv 5521  (class class class)co 6195   ~Hchil 24468   _|_cort 24479    vH chj 24482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-hilex 24548
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-chj 24860
This theorem is referenced by:  shjval  24901  sshjval3  24904  sshjcl  24905  sshjval2  24961  ssjo  24997  sshhococi  25096
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