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Theorem hommval 28444
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 27705 . . 3 ℋ ∈ V
21, 1elmap 7830 . 2 (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
3 oveq1 6611 . . . 4 (𝑓 = 𝐴 → (𝑓 · (𝑔𝑥)) = (𝐴 · (𝑔𝑥)))
43mpteq2dv 4705 . . 3 (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))))
5 fveq1 6147 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
65oveq2d 6620 . . . 4 (𝑔 = 𝑇 → (𝐴 · (𝑔𝑥)) = (𝐴 · (𝑇𝑥)))
76mpteq2dv 4705 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
8 df-homul 28439 . . 3 ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
91mptex 6440 . . 3 (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))) ∈ V
104, 7, 8, 9ovmpt2 6749 . 2 ((𝐴 ∈ ℂ ∧ 𝑇 ∈ ( ℋ ↑𝑚 ℋ)) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
112, 10sylan2br 493 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  cc 9878  chil 27625   · csm 27627   ·op chot 27645
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-hilex 27705
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-homul 28439
This theorem is referenced by:  homval  28449  homulcl  28467
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