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Theorem ocel 28985
Description: Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
ocel (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Distinct variable groups:   𝑥,𝐻   𝑥,𝐴

Proof of Theorem ocel
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ocval 28984 . . 3 (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0})
21eleq2d 2895 . 2 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0}))
3 oveq1 7152 . . . . 5 (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
43eqeq1d 2820 . . . 4 (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
54ralbidv 3194 . . 3 (𝑦 = 𝐴 → (∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0))
65elrab 3677 . 2 (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0))
72, 6syl6bb 288 1 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wss 3933  cfv 6348  (class class class)co 7145  0cc0 10525  chba 28623   ·ih csp 28626  cort 28634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oc 28956
This theorem is referenced by:  shocel  28986  ocsh  28987  ocorth  28995  ococss  28997  occllem  29007  occl  29008  chocnul  29032  h1deoi  29253  h1dei  29254  hmopidmpji  29856
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