Theorem h1deot 5454

Description: Membership in orthocomplement of 1-dimensional subspace.

Hypothesis

Ref	Expression
h1deot.1	⊢ B ∈ ℋ

Assertion

Ref	Expression
h1deot	⊢ (A ∈ (⊥ ‘{B}) ↔ (A ∈ ℋ ∧ (A ·i B) = 0))

Proof of Theorem h1deot

Step	Hyp	Ref	Expression
1		h1deot.1	. . . 4 ⊢ B ∈ ℋ
2		snssi 1851	. . . 4 ⊢ (B ∈ ℋ → {B} ⊆ ℋ )
3	1, 2	ax-mp 6	. . 3 ⊢ {B} ⊆ ℋ
4		ocelt 5162	. . 3 ⊢ ({B} ⊆ ℋ → (A ∈ (⊥ ‘{B}) ↔ (A ∈ ℋ ∧ ∀x ∈ {B} (A ·i x) = 0)))
5	3, 4	ax-mp 6	. 2 ⊢ (A ∈ (⊥ ‘{B}) ↔ (A ∈ ℋ ∧ ∀x ∈ {B} (A ·i x) = 0))
6		df-ral 1205	. . . 4 ⊢ (∀x ∈ {B} (A ·i x) = 0 ↔ ∀x(x ∈ {B} → (A ·i x) = 0))
7		elsn 1820	. . . . . 6 ⊢ (x ∈ {B} ↔ x = B)
8	7	imbi1i 161	. . . . 5 ⊢ ((x ∈ {B} → (A ·i x) = 0) ↔ (x = B → (A ·i x) = 0))
9	8	bial 695	. . . 4 ⊢ (∀x(x ∈ {B} → (A ·i x) = 0) ↔ ∀x(x = B → (A ·i x) = 0))
10	1	elisseti 1355	. . . . 5 ⊢ B ∈ V
11		opreq2 3007	. . . . . 6 ⊢ (x = B → (A ·i x) = (A ·i B))
12	11	cleq1d 1109	. . . . 5 ⊢ (x = B → ((A ·i x) = 0 ↔ (A ·i B) = 0))
13	10, 12	ceqsalv 1364	. . . 4 ⊢ (∀x(x = B → (A ·i x) = 0) ↔ (A ·i B) = 0)
14	6, 9, 13	3bitr 155	. . 3 ⊢ (∀x ∈ {B} (A ·i x) = 0 ↔ (A ·i B) = 0)
15	14	anbi2i 367	. 2 ⊢ ((A ∈ ℋ ∧ ∀x ∈ {B} (A ·i x) = 0) ↔ (A ∈ ℋ ∧ (A ·i B) = 0))
16	5, 15	bitr 151	1 ⊢ (A ∈ (⊥ ‘{B}) ↔ (A ∈ ℋ ∧ (A ·i B) = 0))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  {csn 1808   ‘cfv 2422  (class class class)co 3001  0cc0 4028   ℋ chil 4958   ·_i csp 4963  ⊥cort 4969

This theorem is referenced by:  h1det 5455

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oc 5156

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