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

Theorem bracl 22584
Description: Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
bracl  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )

Proof of Theorem bracl
StepHypRef Expression
1 brafn 22582 . 2  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
2 ffvelrn 5701 . 2  |-  ( ( ( bra `  A
) : ~H --> CC  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
31, 2sylan 457 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1701   -->wf 5288   ` cfv 5292   CCcc 8780   ~Hchil 21554   bracbr 21591
This theorem is referenced by:  kbass2  22752  kbass3  22753  kbass4  22754  kbass6  22756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-hilex 21634  ax-hfi 21713
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-bra 22485
  Copyright terms: Public domain W3C validator