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Mirrors > Home > MPE Home > Th. List > obsip | Structured version Visualization version GIF version |
Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
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) |
Ref | Expression |
---|---|
obsip | ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )) |
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 2725 | . . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
7 | eqid 2725 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 21671 | . . . . 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 1144 | . . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)})) |
10 | 9 | simpld 493 | . . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )) |
11 | oveq1 7426 | . . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦)) | |
12 | eqeq1 2729 | . . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 = 𝑦 ↔ 𝑃 = 𝑦)) | |
13 | 12 | ifbid 4553 | . . . . 5 ⊢ (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 )) |
14 | 11, 13 | eqeq12d 2741 | . . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ))) |
15 | oveq2 7427 | . . . . 5 ⊢ (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄)) | |
16 | eqeq2 2737 | . . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 = 𝑦 ↔ 𝑃 = 𝑄)) | |
17 | 16 | ifbid 4553 | . . . . 5 ⊢ (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 )) |
18 | 15, 17 | eqeq12d 2741 | . . . 4 ⊢ (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))) |
19 | 14, 18 | rspc2v 3617 | . . 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 1113 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 ifcif 4530 {csn 4630 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 ·𝑖cip 17241 0gc0g 17424 1rcur 20133 PreHilcphl 21573 ocvcocv 21609 OBasiscobs 21653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-obs 21656 |
This theorem is referenced by: obsipid 21673 obselocv 21679 |
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