Theorem ocel 28712

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 28711	. . 3 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0})
2	1	eleq2d 2844	. 2 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0}))
3		oveq1 6929	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
4	3	eqeq1d 2779	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
5	4	ralbidv 3167	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
6	5	elrab 3571	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
7	2, 6	syl6bb 279	1 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∀wral 3089  {crab 3093   ⊆ wss 3791  ‘cfv 6135  (class class class)co 6922  0cc0 10272   ℋchba 28348   ·_ih csp 28351  ⊥cort 28359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-hilex 28428

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oc 28681

This theorem is referenced by:  shocel  28713  ocsh  28714  ocorth  28722  ococss  28724  occllem  28734  occl  28735  chocnul  28759  h1deoi  28980  h1dei  28981  hmopidmpji  29583