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Theorem homcl 25069
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 25064 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
2 ffvelrn 5838 . . . . 5  |-  ( ( T : ~H --> ~H  /\  B  e.  ~H )  ->  ( T `  B
)  e.  ~H )
32anim2i 566 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  B  e.  ~H )
)  ->  ( A  e.  CC  /\  ( T `
 B )  e. 
~H ) )
433impb 1178 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( A  e.  CC  /\  ( T `  B
)  e.  ~H )
)
5 hvmulcl 24334 . . 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 2515 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    e. wcel 1761   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   ~Hchil 24240    .h csm 24242    .op chot 24260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-hilex 24320  ax-hfvmul 24326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-homul 25054
This theorem is referenced by: (None)
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