Theorem ocel 29544

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 ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))

Distinct variable groups:   𝑥,𝐻   𝑥,𝐴

Proof of Theorem ocel

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocval 29543	. . 3 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0})
2	1	eleq2d 2824	. 2 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0}))
3		oveq1 7262	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
4	3	eqeq1d 2740	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
5	4	ralbidv 3120	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
6	5	elrab 3617	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
7	2, 6	bitrdi 286	1 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  ‘cfv 6418  (class class class)co 7255  0cc0 10802   ℋchba 29182   ·_ih csp 29185  ⊥cort 29193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oc 29515

This theorem is referenced by:  shocel  29545  ocsh  29546  ocorth  29554  ococss  29556  occllem  29566  occl  29567  chocnul  29591  h1deoi  29812  h1dei  29813  hmopidmpji  30415