Theorem hommval 28904

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 ·ℎ (𝑇‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem hommval

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

Step	Hyp	Ref	Expression
1		ax-hilex 28165	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8052	. 2 ⊢ (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
3		oveq1 6820	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑔‘𝑥)))
4	3	mpteq2dv 4897	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))))
5		fveq1 6351	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
6	5	oveq2d 6829	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑇‘𝑥)))
7	6	mpteq2dv 4897	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
8		df-homul 28899	. . 3 ⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
9	1	mptex 6650	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))) ∈ V
10	4, 7, 8, 9	ovmpt2 6961	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ ( ℋ ↑𝑚 ℋ)) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
11	2, 10	sylan2br 494	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ↦ cmpt 4881  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↑_𝑚 cmap 8023  ℂcc 10126   ℋchil 28085   ·_ℎ csm 28087   ·_op chot 28105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-hilex 28165

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-homul 28899

This theorem is referenced by:  homval  28909  homulcl  28927