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Theorem sshjval3 25948
Description: Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice  CH. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
sshjval3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  (  \/H  `  { A ,  B } ) )

Proof of Theorem sshjval3
StepHypRef Expression
1 ax-hilex 25592 . . . . . 6  |-  ~H  e.  _V
21elpw2 4611 . . . . 5  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4611 . . . . 5  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uniprg 4259 . . . . 5  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  U. { A ,  B }  =  ( A  u.  B )
)
52, 3, 4syl2anbr 480 . . . 4  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  U. { A ,  B }  =  ( A  u.  B ) )
65fveq2d 5868 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  U. { A ,  B } )  =  ( _|_ `  ( A  u.  B )
) )
76fveq2d 5868 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  ( _|_ `  U. { A ,  B }
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
8 prssi 4183 . . . 4  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  { A ,  B }  C_  ~P ~H )
92, 3, 8syl2anbr 480 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  { A ,  B }  C_  ~P ~H )
10 hsupval 25928 . . 3  |-  ( { A ,  B }  C_ 
~P ~H  ->  (  \/H  `  { A ,  B } )  =  ( _|_ `  ( _|_ `  U. { A ,  B } ) ) )
119, 10syl 16 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  (  \/H  `  { A ,  B } )  =  ( _|_ `  ( _|_ `  U. { A ,  B } ) ) )
12 sshjval 25944 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
137, 11, 123eqtr4rd 2519 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  (  \/H  `  { A ,  B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476   ~Pcpw 4010   {cpr 4029   U.cuni 4245   ` cfv 5586  (class class class)co 6282   ~Hchil 25512   _|_cort 25523    vH chj 25526    \/H chsup 25527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-hilex 25592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-chj 25904  df-chsup 25905
This theorem is referenced by: (None)
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