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| Description: Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| obsrcl | ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 4 | eqid 2737 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
| 5 | eqid 2737 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
| 6 | eqid 2737 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
| 7 | eqid 2737 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | isobs 21740 | . 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ (Base‘𝑊) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(·𝑖‘𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)}))) | 
| 9 | 8 | simp1bi 1146 | 1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = 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: obsne0 21745 obs2ocv 21747 obselocv 21748 obslbs 21750 | 
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