Theorem obsocv 22038

Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
obsocv.z	⊢ 0 = (0g‘𝑊)
obsocv.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
obsocv	⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘𝐵) = { 0 })

Proof of Theorem obsocv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2760	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2760	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
4		eqid 2760	. . . 4 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
5		eqid 2760	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6		obsocv.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
7		obsocv.z	. . . 4 ⊢ 0 = (0g‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	isobs 22032	. . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ (Base‘𝑊) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(·𝑖‘𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ( ⊥ ‘𝐵) = { 0 })))
9	8	simp3bi 1165	. 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(·𝑖‘𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ( ⊥ ‘𝐵) = { 0 }))
10	9	simprd 501	1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘𝐵) = { 0 })

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ifcif 4481  {csn 4584  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  Scalarcsca 17437  ·_𝑖cip 17439  0_gc0g 17616  1_rcur 20413  PreHilcphl 21936  ocvcocv 21972  OBasiscobs 22014

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-obs 22017

This theorem is used by:  obs2ocv  22039  obs2ss  22041