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

Theorem shocel 28111
Description: Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shocel (𝐻S → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Distinct variable groups:   𝑥,𝐻   𝑥,𝐴

Proof of Theorem shocel
StepHypRef Expression
1 shss 28037 . 2 (𝐻S𝐻 ⊆ ℋ)
2 ocel 28110 . 2 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
31, 2syl 17 1 (𝐻S → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  wss 3567  cfv 5876  (class class class)co 6635  0cc0 9921  chil 27746   ·ih csp 27749   S csh 27755  cort 27757
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
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-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-br 4645  df-opab 4704  df-mpt 4721  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-fv 5884  df-ov 6638  df-sh 28034  df-oc 28079
This theorem is referenced by:  ocin  28125  choc0  28155  choc1  28156  pjhthlem2  28221  pjclem4  29028  pj3si  29036
  Copyright terms: Public domain W3C validator