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Theorem shaddcl 28599
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 28591 . . . . 5 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
21simprbi 491 . . . 4 (𝐻S → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
32simpld 489 . . 3 (𝐻S → ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻)
4 oveq1 6885 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
54eleq1d 2863 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝐻 ↔ (𝐴 + 𝑦) ∈ 𝐻))
6 oveq2 6886 . . . . 5 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
76eleq1d 2863 . . . 4 (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝐻 ↔ (𝐴 + 𝐵) ∈ 𝐻))
85, 7rspc2v 3510 . . 3 ((𝐴𝐻𝐵𝐻) → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 → (𝐴 + 𝐵) ∈ 𝐻))
93, 8syl5com 31 . 2 (𝐻S → ((𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻))
1093impib 1145 1 ((𝐻S𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wss 3769  (class class class)co 6878  cc 10222  chba 28301   + cva 28302   · csm 28303  0c0v 28306   S csh 28310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-hilex 28381  ax-hfvadd 28382  ax-hfvmul 28387
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-sh 28589
This theorem is referenced by:  shsubcl  28602  hhssabloilem  28643  hhssnv  28646  shscli  28701  shintcli  28713  shsleji  28754  shsidmi  28768  pjhthlem1  28775  spanuni  28928  spanunsni  28963  sumspansn  29033  pjaddii  29059  imaelshi  29442
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