Theorem h1deoi 32208

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 4748	. . 3 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
3		ocel 31940	. . 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 3476	. . . 4 ⊢ 𝐵 ∈ V
6		oveq2 7459	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐴 ·ih 𝐵))
7	6	eqeq1d 2765	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0))
8	5, 7	ralsn 4643	. . 3 ⊢ (∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0)
9	8	anbi2i 635	. 2 ⊢ ((𝐴 ∈ ℋ ∧ ∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))
10	4, 9	bitri 278	1 ⊢ (𝐴 ∈ (⊥‘{𝐵}) ↔ (𝐴 ∈ ℋ ∧ (𝐴 ·ih 𝐵) = 0))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  {csn 4588  ‘cfv 6566  (class class class)co 7451  0cc0 11239   ℋchba 31578   ·_ih csp 31581  ⊥cort 31589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-hilex 31658

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oc 31911

This theorem is used by:  h1dei  32209