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Theorem sshjval 22242
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 21892 . . 3  |-  ~H  e.  _V
21elpw2 4277 . 2  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4277 . 2  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uneq12 3412 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  u.  y
)  =  ( A  u.  B ) )
54fveq2d 5636 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  (
x  u.  y ) )  =  ( _|_ `  ( A  u.  B
) ) )
65fveq2d 5636 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
7 df-chj 22202 . . 3  |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
8 fvex 5646 . . 3  |-  ( _|_ `  ( _|_ `  ( A  u.  B )
) )  e.  _V
96, 7, 8ovmpt2a 6104 . 2  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
102, 3, 9syl2anbr 466 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    u. cun 3236    C_ wss 3238   ~Pcpw 3714   ` cfv 5358  (class class class)co 5981   ~Hchil 21812   _|_cort 21823    vH chj 21826
This theorem is referenced by:  shjval  22243  sshjval3  22246  sshjcl  22247  sshjval2  22303  ssjo  22339  sshhococi  22438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-hilex 21892
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-chj 22202
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