Theorem obsip 22057

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 2763	. . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
7		eqid 2763	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	isobs 22056	. . . . 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 1165	. . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)}))
10	9	simpld 500	. . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11		oveq1 7450	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12		eqeq1 2767	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 = 𝑦 ↔ 𝑃 = 𝑦))
13	12	ifbid 4509	. . . . 5 ⊢ (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
14	11, 13	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15		oveq2 7451	. . . . 5 ⊢ (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16		eqeq2 2775	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 = 𝑦 ↔ 𝑃 = 𝑄))
17	16	ifbid 4509	. . . . 5 ⊢ (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
18	15, 17	eqeq12d 2779	. . . 4 ⊢ (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
19	14, 18	rspc2v 3590	. . 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 32	. 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ((𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
21	20	3impib 1134	1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  ifcif 4485  {csn 4588  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  Scalarcsca 17460  ·_𝑖cip 17462  0_gc0g 17639  1_rcur 20437  PreHilcphl 21960  ocvcocv 21996  OBasiscobs 22038

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 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

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-ne 2960  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 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-obs 22041

This theorem is used by:  obsipid  22058  obselocv  22064