Theorem ocel 31430

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 31429	. . 3 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0})
2	1	eleq2d 2847	. 2 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ 𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0}))
3		oveq1 7399	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ·ih 𝑥) = (𝐴 ·ih 𝑥))
4	3	eqeq1d 2763	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝑥) = 0))
5	4	ralbidv 3184	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
6	5	elrab 3650	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℋ ∣ ∀𝑥 ∈ 𝐻 (𝑦 ·ih 𝑥) = 0} ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))
7	2, 6	bitrdi 289	1 ⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3904  ‘cfv 6517  (class class class)co 7392  0cc0 11070   ℋchba 31068   ·_ih csp 31071  ⊥cort 31079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-hilex 31148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oc 31401

This theorem is referenced by:  shocel  31431  ocsh  31432  ocorth  31440  ococss  31442  occllem  31452  occl  31453  chocnul  31477  h1deoi  31698  h1dei  31699  hmopidmpji  32301