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Theorem hommval 29498
 Description: Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hommval ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem hommval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28761 . . 3 ℋ ∈ V
21, 1elmap 8410 . 2 (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
3 oveq1 7137 . . . 4 (𝑓 = 𝐴 → (𝑓 · (𝑔𝑥)) = (𝐴 · (𝑔𝑥)))
43mpteq2dv 5135 . . 3 (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))))
5 fveq1 6642 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
65oveq2d 7146 . . . 4 (𝑔 = 𝑇 → (𝐴 · (𝑔𝑥)) = (𝐴 · (𝑇𝑥)))
76mpteq2dv 5135 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
8 df-homul 29493 . . 3 ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
91mptex 6959 . . 3 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) ∈ V
104, 7, 8, 9ovmpo 7284 . 2 ((𝐴 ∈ ℂ ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
112, 10sylan2br 597 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ↦ cmpt 5119  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130   ↑m cmap 8381  ℂcc 10512   ℋchba 28681   ·ℎ csm 28683   ·op chot 28701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-hilex 28761 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-homul 29493 This theorem is referenced by:  homval  29503  homulcl  29521
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