Theorem hfmmval 31675

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 11156	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 30935	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8847	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		oveq1 7397	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 · (𝑔‘𝑥)) = (𝐴 · (𝑔‘𝑥)))
5	4	mpteq2dv 5204	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))))
6		fveq1 6860	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
7	6	oveq2d 7406	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 · (𝑔‘𝑥)) = (𝐴 · (𝑇‘𝑥)))
8	7	mpteq2dv 5204	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))
9		df-hfmul 31670	. . 3 ⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
10	2	mptex 7200	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))) ∈ V
11	5, 8, 9, 10	ovmpo 7552	. 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 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  ℂcc 11073   · cmul 11080   ℋchba 30855   ·_fn chft 30878

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-hilex 30935

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-hfmul 31670

This theorem is referenced by:  hfmval  31680  brafnmul  31887  kbass2  32053