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

Theorem homcl 23202
Description: Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homcl  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  e.  ~H )

Proof of Theorem homcl
StepHypRef Expression
1 homval 23197 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
2 ffvelrn 5827 . . . . 5  |-  ( ( T : ~H --> ~H  /\  B  e.  ~H )  ->  ( T `  B
)  e.  ~H )
32anim2i 553 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  B  e.  ~H )
)  ->  ( A  e.  CC  /\  ( T `
 B )  e. 
~H ) )
433impb 1149 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( A  e.  CC  /\  ( T `  B
)  e.  ~H )
)
5 hvmulcl 22469 . . 3  |-  ( ( A  e.  CC  /\  ( T `  B )  e.  ~H )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
64, 5syl 16 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( A  .h  ( T `  B )
)  e.  ~H )
71, 6eqeltrd 2478 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   ~Hchil 22375    .h csm 22377    .op chot 22395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-hilex 22455  ax-hfvmul 22461
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-homul 23187
  Copyright terms: Public domain W3C validator