Theorem h1deoi 30802

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 4812	. . 3 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
3		ocel 30534	. . 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 3494	. . . 4 ⊢ 𝐵 ∈ V
6		oveq2 7417	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐴 ·ih 𝐵))
7	6	eqeq1d 2735	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0))
8	5, 7	ralsn 4686	. . 3 ⊢ (∀𝑥 ∈ {𝐵} (𝐴 ·ih 𝑥) = 0 ↔ (𝐴 ·ih 𝐵) = 0)
9	8	anbi2i 624	. 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 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949  {csn 4629  ‘cfv 6544  (class class class)co 7409  0cc0 11110   ℋchba 30172   ·_ih csp 30175  ⊥cort 30183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-hilex 30252

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oc 30505

This theorem is referenced by:  h1dei  30803