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Theorem hfmmval 29207
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 10464 . . 3 ℂ ∈ V
2 ax-hilex 28467 . . 3 ℋ ∈ V
31, 2elmap 8285 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 oveq1 7023 . . . 4 (𝑓 = 𝐴 → (𝑓 · (𝑔𝑥)) = (𝐴 · (𝑔𝑥)))
54mpteq2dv 5056 . . 3 (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))))
6 fveq1 6537 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
76oveq2d 7032 . . . 4 (𝑔 = 𝑇 → (𝐴 · (𝑔𝑥)) = (𝐴 · (𝑇𝑥)))
87mpteq2dv 5056 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
9 df-hfmul 29202 . . 3 ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
102mptex 6852 . . 3 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) ∈ V
115, 8, 9, 10ovmpo 7166 . 2 ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (ℂ ↑𝑚 ℋ)) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
123, 11sylan2br 594 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  cmpt 5041  wf 6221  cfv 6225  (class class class)co 7016  𝑚 cmap 8256  cc 10381   · cmul 10388  chba 28387   ·fn chft 28410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-hilex 28467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-map 8258  df-hfmul 29202
This theorem is referenced by:  hfmval  29212  brafnmul  29419  kbass2  29585
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