HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hsupval Structured version   Unicode version

Theorem hsupval 22828
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 22903. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hsupval  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )

Proof of Theorem hsupval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 22494 . . . 4  |-  ~H  e.  _V
21pwex 4374 . . 3  |-  ~P ~H  e.  _V
32elpw2 4356 . 2  |-  ( A  e.  ~P ~P ~H  <->  A 
C_  ~P ~H )
4 unieq 4016 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
54fveq2d 5724 . . . 4  |-  ( x  =  A  ->  ( _|_ `  U. x )  =  ( _|_ `  U. A ) )
65fveq2d 5724 . . 3  |-  ( x  =  A  ->  ( _|_ `  ( _|_ `  U. x ) )  =  ( _|_ `  ( _|_ `  U. A ) ) )
7 df-chsup 22805 . . 3  |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
8 fvex 5734 . . 3  |-  ( _|_ `  ( _|_ `  U. A ) )  e. 
_V
96, 7, 8fvmpt 5798 . 2  |-  ( A  e.  ~P ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
103, 9sylbir 205 1  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   ` cfv 5446   ~Hchil 22414   _|_cort 22425    \/H chsup 22429
This theorem is referenced by:  chsupval  22829  hsupcl  22833  hsupss  22835  hsupunss  22837  sshjval3  22848  hsupval2  22903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-chsup 22805
  Copyright terms: Public domain W3C validator