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Theorem shaddcl 28993
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 28985 . . . . 5 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
21simprbi 499 . . . 4 (𝐻S → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
32simpld 497 . . 3 (𝐻S → ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻)
4 oveq1 7162 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
54eleq1d 2897 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝐻 ↔ (𝐴 + 𝑦) ∈ 𝐻))
6 oveq2 7163 . . . . 5 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
76eleq1d 2897 . . . 4 (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝐻 ↔ (𝐴 + 𝐵) ∈ 𝐻))
85, 7rspc2v 3632 . . 3 ((𝐴𝐻𝐵𝐻) → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 → (𝐴 + 𝐵) ∈ 𝐻))
93, 8syl5com 31 . 2 (𝐻S → ((𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻))
1093impib 1112 1 ((𝐻S𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  wss 3935  (class class class)co 7155  cc 10534  chba 28695   + cva 28696   · csm 28697  0c0v 28700   S csh 28704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-hilex 28775  ax-hfvadd 28776  ax-hfvmul 28781
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-sh 28983
This theorem is referenced by:  shsubcl  28996  hhssabloilem  29037  hhssnv  29040  shscli  29093  shintcli  29105  shsleji  29146  shsidmi  29160  pjhthlem1  29167  spanuni  29320  spanunsni  29355  sumspansn  29425  pjaddii  29451  imaelshi  29834
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