Theorem hfmmval 31725

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 11215	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 30985	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8890	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		oveq1 7417	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 · (𝑔‘𝑥)) = (𝐴 · (𝑔‘𝑥)))
5	4	mpteq2dv 5220	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))))
6		fveq1 6880	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
7	6	oveq2d 7426	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 · (𝑔‘𝑥)) = (𝐴 · (𝑇‘𝑥)))
8	7	mpteq2dv 5220	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))
9		df-hfmul 31720	. . 3 ⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
10	2	mptex 7220	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))) ∈ V
11	5, 8, 9, 10	ovmpo 7572	. 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 5206  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845  ℂcc 11132   · cmul 11139   ℋchba 30905   ·_fn chft 30928

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-hilex 30985

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-hfmul 31720

This theorem is referenced by:  hfmval  31730  brafnmul  31937  kbass2  32103