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Theorem homval 28488
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 28483 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
21fveq1d 6160 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))‘𝐵))
3 fveq2 6158 . . . . 5 (𝑥 = 𝐵 → (𝑇𝑥) = (𝑇𝐵))
43oveq2d 6631 . . . 4 (𝑥 = 𝐵 → (𝐴 · (𝑇𝑥)) = (𝐴 · (𝑇𝐵)))
5 eqid 2621 . . . 4 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))
6 ovex 6643 . . . 4 (𝐴 · (𝑇𝐵)) ∈ V
74, 5, 6fvmpt 6249 . . 3 (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥)))‘𝐵) = (𝐴 · (𝑇𝐵)))
82, 7sylan9eq 2675 . 2 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))
983impa 1256 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 384  w3a 1036   = wceq 1480  wcel 1987  cmpt 4683  wf 5853  cfv 5857  (class class class)co 6615  cc 9894  chil 27664   · csm 27666   ·op chot 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-hilex 27744
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-homul 28478
This theorem is referenced by:  homcl  28493  honegsubi  28543  homulid2  28547  homco1  28548  homulass  28549  hoadddi  28550  hoadddir  28551  nmopnegi  28712  homco2  28724  lnopmi  28747  hmopm  28768  nmophmi  28778  adjmul  28839  leopmuli  28880  leopnmid  28885
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