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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  ℂcc 10146   ℋchil 28106   ·ℎ 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
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