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Theorem sshjval3 21858
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 21504 . . . . . 6  |-  ~H  e.  _V
21elpw2 4108 . . . . 5  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4108 . . . . 5  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uniprg 3783 . . . . 5  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  U. { A ,  B }  =  ( A  u.  B )
)
52, 3, 4syl2anbr 468 . . . 4  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  U. { A ,  B }  =  ( A  u.  B ) )
65fveq2d 5427 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  U. { A ,  B } )  =  ( _|_ `  ( A  u.  B )
) )
76fveq2d 5427 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  ( _|_ `  U. { A ,  B }
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
8 prssi 3712 . . . 4  |-  ( ( A  e.  ~P ~H  /\  B  e.  ~P ~H )  ->  { A ,  B }  C_  ~P ~H )
92, 3, 8syl2anbr 468 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  { A ,  B }  C_  ~P ~H )
10 hsupval 21838 . . 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 21854 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
137, 11, 123eqtr4rd 2299 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  (  \/H  `  { A ,  B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    u. cun 3092    C_ wss 3094   ~Pcpw 3566   {cpr 3582   U.cuni 3768   ` cfv 4638  (class class class)co 5757   ~Hchil 21424   _|_cort 21435    vH chj 21438    \/H chsup 21439
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-hilex 21504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-chj 21814  df-chsup 21815
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