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Theorem sshjval3 21933
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 21579 . . . . . 6  |-  ~H  e.  _V
21elpw2 4175 . . . . 5  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4175 . . . . 5  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uniprg 3842 . . . . 5  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  U. { A ,  B }  =  ( A  u.  B )
)
52, 3, 4syl2anbr 466 . . . 4  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  U. { A ,  B }  =  ( A  u.  B ) )
65fveq2d 5529 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  U. { A ,  B } )  =  ( _|_ `  ( A  u.  B )
) )
76fveq2d 5529 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  ( _|_ `  U. { A ,  B }
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
8 prssi 3771 . . . 4  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  { A ,  B }  C_  ~P ~H )
92, 3, 8syl2anbr 466 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  { A ,  B }  C_  ~P ~H )
10 hsupval 21913 . . 3  |-  ( { A ,  B }  C_ 
~P ~H  ->  (  \/H  `  { A ,  B } )  =  ( _|_ `  ( _|_ `  U. { A ,  B } ) ) )
119, 10syl 15 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  (  \/H  `  { A ,  B } )  =  ( _|_ `  ( _|_ `  U. { A ,  B } ) ) )
12 sshjval 21929 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
137, 11, 123eqtr4rd 2326 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  (  \/H  `  { A ,  B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   ~Pcpw 3625   {cpr 3641   U.cuni 3827   ` cfv 5255  (class class class)co 5858   ~Hchil 21499   _|_cort 21510    vH chj 21513    \/H chsup 21514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-chj 21889  df-chsup 21890
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