Theorem hommval 32225

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 31488	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8882	. 2 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
3		oveq1 7424	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑔‘𝑥)))
4	3	mpteq2dv 5203	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))))
5		fveq1 6881	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
6	5	oveq2d 7433	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑇‘𝑥)))
7	6	mpteq2dv 5203	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
8		df-homul 32220	. . 3 ⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
9	1	mptex 7226	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))) ∈ V
10	4, 7, 8, 9	ovmpo 7577	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
11	2, 10	sylan2br 607	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5190  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830  ℂcc 11126   ℋchba 31408   ·_ℎ csm 31410   ·_op chot 31428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-hilex 31488

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8832  df-homul 32220

This theorem is used by:  homval  32230  homulcl  32248