Theorem hommval 31413

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 30676	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8860	. 2 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
3		oveq1 7408	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑔‘𝑥)))
4	3	mpteq2dv 5240	. . 3 ⊢ (𝑓 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))))
5		fveq1 6880	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
6	5	oveq2d 7417	. . . 4 ⊢ (𝑔 = 𝑇 → (𝐴 ·ℎ (𝑔‘𝑥)) = (𝐴 ·ℎ (𝑇‘𝑥)))
7	6	mpteq2dv 5240	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
8		df-homul 31408	. . 3 ⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
9	1	mptex 7216	. . 3 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))) ∈ V
10	4, 7, 8, 9	ovmpo 7560	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))
11	2, 10	sylan2br 594	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5221  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401   ↑_m cmap 8815  ℂcc 11103   ℋchba 30596   ·_ℎ csm 30598   ·_op chot 30616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-hilex 30676

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-homul 31408

This theorem is referenced by:  homval  31418  homulcl  31436