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Theorem sshjval3 21894
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 21540 . . . . . 6  |-  ~H  e.  _V
21elpw2 4142 . . . . 5  |-  ( A  e.  ~P ~H  <->  A  C_  ~H )
31elpw2 4142 . . . . 5  |-  ( B  e.  ~P ~H  <->  B  C_  ~H )
4 uniprg 3816 . . . . 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 5462 . . 3  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  U. { A ,  B } )  =  ( _|_ `  ( A  u.  B )
) )
76fveq2d 5462 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( _|_ `  ( _|_ `  U. { A ,  B }
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) ) )
8 prssi 3745 . . . 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 21874 . . 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 21890 . 2  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
137, 11, 123eqtr4rd 2301 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 3125    C_ wss 3127   ~Pcpw 3599   {cpr 3615   U.cuni 3801   ` cfv 4673  (class class class)co 5792   ~Hchil 21460   _|_cort 21471    vH chj 21474    \/H chsup 21475
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-hilex 21540
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-chj 21850  df-chsup 21851
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