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Theorem ocel 31300
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 31299 . . 3 (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0})
21eleq2d 2827 . 2 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0}))
3 oveq1 7438 . . . . 5 (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
43eqeq1d 2739 . . . 4 (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
54ralbidv 3178 . . 3 (𝑦 = 𝐴 → (∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0))
65elrab 3692 . 2 (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0))
72, 6bitrdi 287 1 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951  cfv 6561  (class class class)co 7431  0cc0 11155  chba 30938   ·ih csp 30941  cort 30949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oc 31271
This theorem is referenced by:  shocel  31301  ocsh  31302  ocorth  31310  ococss  31312  occllem  31322  occl  31323  chocnul  31347  h1deoi  31568  h1dei  31569  hmopidmpji  32171
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