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

Theorem shaddcl 27458
 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 27450 . . . . 5 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
21simprbi 479 . . . 4 (𝐻S → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
32simpld 474 . . 3 (𝐻S → ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻)
4 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
54eleq1d 2672 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝐻 ↔ (𝐴 + 𝑦) ∈ 𝐻))
6 oveq2 6557 . . . . 5 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
76eleq1d 2672 . . . 4 (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝐻 ↔ (𝐴 + 𝐵) ∈ 𝐻))
85, 7rspc2v 3293 . . 3 ((𝐴𝐻𝐵𝐻) → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 → (𝐴 + 𝐵) ∈ 𝐻))
93, 8syl5com 31 . 2 (𝐻S → ((𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻))
1093impib 1254 1 ((𝐻S𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  (class class class)co 6549  ℂcc 9813   ℋchil 27160   +ℎ cva 27161   ·ℎ csm 27162  0ℎc0v 27165   Sℋ csh 27169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-hilex 27240  ax-hfvadd 27241  ax-hfvmul 27246 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-sh 27448 This theorem is referenced by:  shsubcl  27461  hhssabloilem  27502  hhssnv  27505  shscli  27560  shintcli  27572  shsleji  27613  shsidmi  27627  pjhthlem1  27634  spanuni  27787  spanunsni  27822  sumspansn  27892  pjaddii  27918  imaelshi  28301
 Copyright terms: Public domain W3C validator