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

Theorem shaddcl 31236
Description: Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shaddcl ((𝐻S𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻)

Proof of Theorem shaddcl
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))
32simpld 494 . . 3 (𝐻S → ∀𝑥𝐻𝑦𝐻 (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 ((𝐴𝐻𝐵𝐻) → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 → (𝐴 + 𝐵) ∈ 𝐻))
93, 8syl5com 31 . 2 (𝐻S → ((𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻))
1093impib 1117 1 ((𝐻S𝐴𝐻𝐵𝐻) → (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  hhssabloilem  31280  hhssnv  31283  shscli  31336  shintcli  31348  shsleji  31389  shsidmi  31403  pjhthlem1  31410  spanuni  31563  spanunsni  31598  sumspansn  31668  pjaddii  31694  imaelshi  32077
  Copyright terms: Public domain W3C validator