Theorem hfmmval 31668

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 · (𝑇‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem hfmmval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 11149	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 30928	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8844	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		oveq1 7394	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 · (𝑔‘𝑥)) = (𝐴 · (𝑔‘𝑥)))
5	4	mpteq2dv 5201	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))))
6		fveq1 6857	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
7	6	oveq2d 7403	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 · (𝑔‘𝑥)) = (𝐴 · (𝑇‘𝑥)))
8	7	mpteq2dv 5201	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))
9		df-hfmul 31663	. . 3 ⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
10	2	mptex 7197	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))) ∈ V
11	5, 8, 9, 10	ovmpo 7549	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (ℂ ↑m ℋ)) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))
12	3, 11	sylan2br 595	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5188  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  ℂcc 11066   · cmul 11073   ℋchba 30848   ·_fn chft 30871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-hilex 30928

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-hfmul 31663

This theorem is referenced by:  hfmval  31673  brafnmul  31880  kbass2  32046