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Theorem hfmmval 31768
Description: Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hfmmval ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem hfmmval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 11234 . . 3 ℂ ∈ V
2 ax-hilex 31028 . . 3 ℋ ∈ V
31, 2elmap 8910 . 2 (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 oveq1 7438 . . . 4 (𝑓 = 𝐴 → (𝑓 · (𝑔𝑥)) = (𝐴 · (𝑔𝑥)))
54mpteq2dv 5250 . . 3 (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))))
6 fveq1 6906 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
76oveq2d 7447 . . . 4 (𝑔 = 𝑇 → (𝐴 · (𝑔𝑥)) = (𝐴 · (𝑇𝑥)))
87mpteq2dv 5250 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
9 df-hfmul 31763 . . 3 ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
102mptex 7243 . . 3 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) ∈ V
115, 8, 9, 10ovmpo 7593 . 2 ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (ℂ ↑m ℋ)) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
123, 11sylan2br 595 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cc 11151   · cmul 11158  chba 30948   ·fn chft 30971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-hfmul 31763
This theorem is referenced by:  hfmval  31773  brafnmul  31980  kbass2  32146
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