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

Theorem shmulcl 31237
Description: Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shmulcl ((𝐻S𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻)

Proof of Theorem shmulcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issh2 31228 . . . . 5 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
21simprbi 496 . . . 4 (𝐻S → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
32simprd 495 . . 3 (𝐻S → ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)
4 oveq1 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
54eleq1d 2826 . . . 4 (𝑥 = 𝐴 → ((𝑥 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝑦) ∈ 𝐻))
6 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
76eleq1d 2826 . . . 4 (𝑦 = 𝐵 → ((𝐴 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝐵) ∈ 𝐻))
85, 7rspc2v 3633 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵𝐻) → (∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻 → (𝐴 · 𝐵) ∈ 𝐻))
93, 8syl5com 31 . 2 (𝐻S → ((𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻))
1093impib 1117 1 ((𝐻S𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951  (class class class)co 7431  cc 11153  chba 30938   + cva 30939   · csm 30940  0c0v 30943   S csh 30947
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-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018  ax-hfvadd 31019  ax-hfvmul 31024
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-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-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  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-fv 6569  df-ov 7434  df-sh 31226
This theorem is referenced by:  shsubcl  31239  norm1exi  31269  hhssabloilem  31280  hhssnv  31283  shsel3  31334  shscli  31336  shintcli  31348  pjhthlem1  31410  h1de2bi  31573  h1de2ctlem  31574  spansni  31576  spansnmul  31583  spansnss  31590  spanunsni  31598  h1datomi  31600  pjmulii  31696  mayete3i  31747  imaelshi  32077  strlem1  32269  cdj1i  32452  cdj3lem1  32453
  Copyright terms: Public domain W3C validator