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Theorem homcl 22326
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 22321 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
2 ffvelrn 5663 . . . . 5  |-  ( ( T : ~H --> ~H  /\  B  e.  ~H )  ->  ( T `  B
)  e.  ~H )
32anim2i 552 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  B  e.  ~H )
)  ->  ( A  e.  CC  /\  ( T `
 B )  e. 
~H ) )
433impb 1147 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( A  e.  CC  /\  ( T `  B
)  e.  ~H )
)
5 hvmulcl 21593 . . 3  |-  ( ( A  e.  CC  /\  ( T `  B )  e.  ~H )  -> 
( A  .h  ( T `  B )
)  e.  ~H )
64, 5syl 15 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( A  .h  ( T `  B )
)  e.  ~H )
71, 6eqeltrd 2357 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 358    /\ w3a 934    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    .h csm 21501    .op chot 21519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579  ax-hfvmul 21585
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-homul 22311
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