HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  homval Structured version   Visualization version   GIF version

Theorem homval 28930
Description: Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
homval ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))

Proof of Theorem homval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hommval 28925 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
21fveq1d 6355 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))‘𝐵))
3 fveq2 6353 . . . . 5 (𝑥 = 𝐵 → (𝑇𝑥) = (𝑇𝐵))
43oveq2d 6830 . . . 4 (𝑥 = 𝐵 → (𝐴 · (𝑇𝑥)) = (𝐴 · (𝑇𝐵)))
5 eqid 2760 . . . 4 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))
6 ovex 6842 . . . 4 (𝐴 · (𝑇𝐵)) ∈ V
74, 5, 6fvmpt 6445 . . 3 (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))‘𝐵) = (𝐴 · (𝑇𝐵)))
82, 7sylan9eq 2814 . 2 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))
983impa 1101 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6814  cc 10146  chil 28106   · csm 28108   ·op chot 28126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-hilex 28186
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-homul 28920
This theorem is referenced by:  homcl  28935  honegsubi  28985  homulid2  28989  homco1  28990  homulass  28991  hoadddi  28992  hoadddir  28993  nmopnegi  29154  homco2  29166  lnopmi  29189  hmopm  29210  nmophmi  29220  adjmul  29281  leopmuli  29322  leopnmid  29327
  Copyright terms: Public domain W3C validator