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Theorem shmulcl 28045
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 28036 . . . . 5 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
21simprbi 480 . . . 4 (𝐻S → (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
32simprd 479 . . 3 (𝐻S → ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)
4 oveq1 6642 . . . . 5 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
54eleq1d 2684 . . . 4 (𝑥 = 𝐴 → ((𝑥 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝑦) ∈ 𝐻))
6 oveq2 6643 . . . . 5 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
76eleq1d 2684 . . . 4 (𝑦 = 𝐵 → ((𝐴 · 𝑦) ∈ 𝐻 ↔ (𝐴 · 𝐵) ∈ 𝐻))
85, 7rspc2v 3317 . . 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 1260 1 ((𝐻S𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wss 3567  (class class class)co 6635  cc 9919  chil 27746   + cva 27747   · csm 27748  0c0v 27751   S csh 27755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-hilex 27826  ax-hfvadd 27827  ax-hfvmul 27832
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-sh 28034
This theorem is referenced by:  shsubcl  28047  norm1exi  28077  hhssabloilem  28088  hhssnv  28091  shsel3  28144  shscli  28146  shintcli  28158  pjhthlem1  28220  h1de2bi  28383  h1de2ctlem  28384  spansni  28386  spansnmul  28393  spansnss  28400  spanunsni  28408  h1datomi  28410  pjmulii  28506  mayete3i  28557  imaelshi  28887  strlem1  29079  cdj1i  29262  cdj3lem1  29263
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