Theorem obsip 20928

Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
isobs.v	⊢ 𝑉 = (Base‘𝑊)
isobs.h	⊢ , = (·𝑖‘𝑊)
isobs.f	⊢ 𝐹 = (Scalar‘𝑊)
isobs.u	⊢ 1 = (1r‘𝐹)
isobs.z	⊢ 0 = (0g‘𝐹)

Assertion

Ref	Expression
obsip	⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

Proof of Theorem obsip

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

Step	Hyp	Ref	Expression
1		isobs.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
2		isobs.h	. . . . . 6 ⊢ , = (·𝑖‘𝑊)
3		isobs.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
4		isobs.u	. . . . . 6 ⊢ 1 = (1r‘𝐹)
5		isobs.z	. . . . . 6 ⊢ 0 = (0g‘𝐹)
6		eqid 2738	. . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
7		eqid 2738	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	isobs 20927	. . . . 5 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)})))
9	8	simp3bi 1146	. . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)}))
10	9	simpld 495	. . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11		oveq1 7282	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12		eqeq1 2742	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 = 𝑦 ↔ 𝑃 = 𝑦))
13	12	ifbid 4482	. . . . 5 ⊢ (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
14	11, 13	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15		oveq2 7283	. . . . 5 ⊢ (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16		eqeq2 2750	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 = 𝑦 ↔ 𝑃 = 𝑄))
17	16	ifbid 4482	. . . . 5 ⊢ (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
18	15, 17	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
19	14, 18	rspc2v 3570	. . 3 ⊢ ((𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
20	10, 19	syl5com 31	. 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ((𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
21	20	3impib 1115	1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ifcif 4459  {csn 4561  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Scalarcsca 16965  ·_𝑖cip 16967  0_gc0g 17150  1_rcur 19737  PreHilcphl 20829  ocvcocv 20865  OBasiscobs 20909

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-obs 20912

This theorem is referenced by:  obsipid  20929  obselocv  20935