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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 2737 | . . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
| 7 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | isobs 21740 | . . . . 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 7438 | . . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦)) | |
| 12 | eqeq1 2741 | . . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 = 𝑦 ↔ 𝑃 = 𝑦)) | |
| 13 | 12 | ifbid 4549 | . . . . 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 7439 | . . . . 5 ⊢ (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄)) | |
| 16 | eqeq2 2749 | . . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 = 𝑦 ↔ 𝑃 = 𝑄)) | |
| 17 | 16 | ifbid 4549 | . . . . 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 3633 | . . 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 )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ifcif 4525 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑖cip 17302 0gc0g 17484 1rcur 20178 PreHilcphl 21642 ocvcocv 21678 OBasiscobs 21722 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-obs 21725 | 
| This theorem is referenced by: obsipid 21742 obselocv 21748 | 
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