Theorem ocel 30816

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 30815	. . 3 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0})
2	1	eleq2d 2818	. 2 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0}))
3		oveq1 7419	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
4	3	eqeq1d 2733	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
5	4	ralbidv 3176	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
6	5	elrab 3683	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
7	2, 6	bitrdi 287	1 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431   ⊆ wss 3948  ‘cfv 6543  (class class class)co 7412  0cc0 11116   ℋchba 30454   ·_ih csp 30457  ⊥cort 30465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-hilex 30534

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oc 30787

This theorem is referenced by:  shocel  30817  ocsh  30818  ocorth  30826  ococss  30828  occllem  30838  occl  30839  chocnul  30863  h1deoi  31084  h1dei  31085  hmopidmpji  31687