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Theorem sshjval3 26999
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 26644 . . . . . 6  |-  ~H  e.  _V
21elpw2 4586 . . . . 5  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4586 . . . . 5  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uniprg 4231 . . . . 5  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  U. { A ,  B }  =  ( A  u.  B )
)
52, 3, 4syl2anbr 483 . . . 4  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  U. { A ,  B }  =  ( A  u.  B ) )
65fveq2d 5883 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  U. { A ,  B } )  =  ( _|_ `  ( A  u.  B )
) )
76fveq2d 5883 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  ( _|_ `  U. { A ,  B }
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
8 prssi 4154 . . . 4  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  { A ,  B }  C_  ~P ~H )
92, 3, 8syl2anbr 483 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  { A ,  B }  C_  ~P ~H )
10 hsupval 26979 . . 3  |-  ( { A ,  B }  C_ 
~P ~H  ->  (  \/H  `  { A ,  B } )  =  ( _|_ `  ( _|_ `  U. { A ,  B } ) ) )
119, 10syl 17 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  (  \/H  `  { A ,  B } )  =  ( _|_ `  ( _|_ `  U. { A ,  B } ) ) )
12 sshjval 26995 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
137, 11, 123eqtr4rd 2475 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  (  \/H  `  { A ,  B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869    u. cun 3435    C_ wss 3437   ~Pcpw 3980   {cpr 3999   U.cuni 4217   ` cfv 5599  (class class class)co 6303   ~Hchil 26564   _|_cort 26575    vH chj 26578    \/H chsup 26579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-hilex 26644
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-chj 26955  df-chsup 26956
This theorem is referenced by: (None)
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