Theorem h1deoi 31307

Description: Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)

Hypothesis

Ref	Expression
h1deot.1	⊢ 𝐵 ∈ ℋ

Assertion

Ref	Expression
h1deoi	⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))

Proof of Theorem h1deoi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		h1deot.1	. . 3 ⊢ 𝐵 ∈ ℋ
2		snssi 4806	. . 3 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
3		ocel 31039	. . 3 ⊢ ({𝐵} ⊆ ℋ → (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0)))
4	1, 2, 3	mp2b 10	. 2 ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0))
5	1	elexi 3488	. . . 4 ⊢ 𝐵 ∈ V
6		oveq2 7412	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐴 ·ih 𝐵))
7	6	eqeq1d 2728	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0))
8	5, 7	ralsn 4680	. . 3 ⊢ (∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0)
9	8	anbi2i 622	. 2 ⊢ ((𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))
10	4, 9	bitri 275	1 ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055   ⊆ wss 3943  {csn 4623  ‘cfv 6536  (class class class)co 7404  0cc0 11109   ℋchba 30677   ·_ih csp 30680  ⊥cort 30688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-hilex 30757

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oc 31010

This theorem is referenced by:  h1dei  31308