Theorem hfmmval 29510

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 10612	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 28770	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8429	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4		oveq1 7157	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 · (𝑔‘𝑥)) = (𝐴 · (𝑔‘𝑥)))
5	4	mpteq2dv 5154	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))))
6		fveq1 6663	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
7	6	oveq2d 7166	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 · (𝑔‘𝑥)) = (𝐴 · (𝑇‘𝑥)))
8	7	mpteq2dv 5154	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 · (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))
9		df-hfmul 29505	. . 3 ⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
10	2	mptex 6980	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))) ∈ V
11	5, 8, 9, 10	ovmpo 7304	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (ℂ ↑m ℋ)) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))
12	3, 11	sylan2br 596	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ↦ cmpt 5138  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ↑_m cmap 8400  ℂcc 10529   · cmul 10536   ℋchba 28690   ·_fn chft 28713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-hilex 28770

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-hfmul 29505

This theorem is referenced by:  hfmval  29515  brafnmul  29722  kbass2  29888