Theorem obsip 21711

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 2737	. . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
7		eqid 2737	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	isobs 21710	. . . . 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 1148	. . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)}))
10	9	simpld 494	. . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ))
11		oveq1 7367	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦))
12		eqeq1 2741	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 = 𝑦 ↔ 𝑃 = 𝑦))
13	12	ifbid 4491	. . . . 5 ⊢ (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 ))
14	11, 13	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 )))
15		oveq2 7368	. . . . 5 ⊢ (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄))
16		eqeq2 2749	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 = 𝑦 ↔ 𝑃 = 𝑄))
17	16	ifbid 4491	. . . . 5 ⊢ (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 ))
18	15, 17	eqeq12d 2753	. . . 4 ⊢ (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )))
19	14, 18	rspc2v 3576	. . 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 1117	1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ifcif 4467  {csn 4568  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Scalarcsca 17214  ·_𝑖cip 17216  0_gc0g 17393  1_rcur 20153  PreHilcphl 21614  ocvcocv 21650  OBasiscobs 21692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-obs 21695

This theorem is referenced by:  obsipid  21712  obselocv  21718