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Theorem shmulcl 28987
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 28978 . . . . 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))
32simprd 498 . . 3 (𝐻S → ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)
4 oveq1 7155 . . . . 5 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
54eleq1d 2895 . . . 4 (𝑥 = 𝐴 → ((𝑥 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝑦) ∈ 𝐻))
6 oveq2 7156 . . . . 5 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
76eleq1d 2895 . . . 4 (𝑦 = 𝐵 → ((𝐴 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝐵) ∈ 𝐻))
85, 7rspc2v 3631 . . 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 1110 1 ((𝐻S𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  wss 3934  (class class class)co 7148  cc 10527  chba 28688   + cva 28689   · csm 28690  0c0v 28693   S csh 28697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-hilex 28768  ax-hfvadd 28769  ax-hfvmul 28774
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-sh 28976
This theorem is referenced by:  shsubcl  28989  norm1exi  29019  hhssabloilem  29030  hhssnv  29033  shsel3  29084  shscli  29086  shintcli  29098  pjhthlem1  29160  h1de2bi  29323  h1de2ctlem  29324  spansni  29326  spansnmul  29333  spansnss  29340  spanunsni  29348  h1datomi  29350  pjmulii  29446  mayete3i  29497  imaelshi  29827  strlem1  30019  cdj1i  30202  cdj3lem1  30203
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