HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  kbval Structured version   Visualization version   GIF version

Theorem kbval 31973
Description: The value of the operator resulting from the outer product 𝐴 𝐵 of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
kbval ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) · 𝐴))

Proof of Theorem kbval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 kbfval 31971 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))
21fveq1d 6908 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴))‘𝐶))
3 oveq1 7438 . . . . 5 (𝑥 = 𝐶 → (𝑥 ·ih 𝐵) = (𝐶 ·ih 𝐵))
43oveq1d 7446 . . . 4 (𝑥 = 𝐶 → ((𝑥 ·ih 𝐵) · 𝐴) = ((𝐶 ·ih 𝐵) · 𝐴))
5 eqid 2737 . . . 4 (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴))
6 ovex 7464 . . . 4 ((𝐶 ·ih 𝐵) · 𝐴) ∈ V
74, 5, 6fvmpt 7016 . . 3 (𝐶 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴))‘𝐶) = ((𝐶 ·ih 𝐵) · 𝐴))
82, 7sylan9eq 2797 . 2 (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) · 𝐴))
983impa 1110 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561  (class class class)co 7431  chba 30938   · csm 30940   ·ih csp 30941   ketbra ck 30976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-kb 31870
This theorem is referenced by:  kbpj  31975  kbass1  32135  kbass2  32136  kbass5  32139  kbass6  32140
  Copyright terms: Public domain W3C validator