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Mirrors > Home > MPE Home > Th. List > obsrcl | Structured version Visualization version GIF version |
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 2736 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2736 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2736 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2736 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
5 | eqid 2736 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | eqid 2736 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
7 | eqid 2736 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 20636 | . 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 1147 | 1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 ifcif 4425 {csn 4527 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Scalarcsca 16752 ·𝑖cip 16754 0gc0g 16898 1rcur 19470 PreHilcphl 20540 ocvcocv 20576 OBasiscobs 20618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-obs 20621 |
This theorem is referenced by: obsne0 20641 obs2ocv 20643 obselocv 20644 obslbs 20646 |
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