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

Theorem shmulcl 31246
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 31237 . . . . 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 7437 . . . . 5 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
54eleq1d 2823 . . . 4 (𝑥 = 𝐴 → ((𝑥 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝑦) ∈ 𝐻))
6 oveq2 7438 . . . . 5 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
76eleq1d 2823 . . . 4 (𝑦 = 𝐵 → ((𝐴 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝐵) ∈ 𝐻))
85, 7rspc2v 3632 . . 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 1115 1 ((𝐻S𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wss 3962  (class class class)co 7430  cc 11150  chba 30947   + cva 30948   · csm 30949  0c0v 30952   S csh 30956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-hilex 31027  ax-hfvadd 31028  ax-hfvmul 31033
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-sh 31235
This theorem is referenced by:  shsubcl  31248  norm1exi  31278  hhssabloilem  31289  hhssnv  31292  shsel3  31343  shscli  31345  shintcli  31357  pjhthlem1  31419  h1de2bi  31582  h1de2ctlem  31583  spansni  31585  spansnmul  31592  spansnss  31599  spanunsni  31607  h1datomi  31609  pjmulii  31705  mayete3i  31756  imaelshi  32086  strlem1  32278  cdj1i  32461  cdj3lem1  32462
  Copyright terms: Public domain W3C validator