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Theorem ocel 31111
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 31110 . . 3 (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0})
21eleq2d 2815 . 2 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0}))
3 oveq1 7433 . . . . 5 (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
43eqeq1d 2730 . . . 4 (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
54ralbidv 3175 . . 3 (𝑦 = 𝐴 → (∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0))
65elrab 3684 . 2 (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0))
72, 6bitrdi 286 1 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  {crab 3430  wss 3949  cfv 6553  (class class class)co 7426  0cc0 11146  chba 30749   ·ih csp 30752  cort 30760
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-hilex 30829
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oc 31082
This theorem is referenced by:  shocel  31112  ocsh  31113  ocorth  31121  ococss  31123  occllem  31133  occl  31134  chocnul  31158  h1deoi  31379  h1dei  31380  hmopidmpji  31982
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