Theorem hommval 29120

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 28381	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8124	. 2 ⊢ (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
3		oveq1 6885	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑔‘𝑥)))
4	3	mpteq2dv 4938	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))))
5		fveq1 6410	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
6	5	oveq2d 6894	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑇‘𝑥)))
7	6	mpteq2dv 4938	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
8		df-homul 29115	. . 3 ⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
9	1	mptex 6715	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))) ∈ V
10	4, 7, 8, 9	ovmpt2 7030	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ ( ℋ ↑𝑚 ℋ)) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
11	2, 10	sylan2br 589	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ↦ cmpt 4922  ⟶wf 6097  ‘cfv 6101  (class class class)co 6878   ↑_𝑚 cmap 8095  ℂcc 10222   ℋchba 28301   ·_ℎ csm 28303   ·_op chot 28321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-hilex 28381

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-homul 29115

This theorem is referenced by:  homval  29125  homulcl  29143