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Mirrors > Home > MPE Home > Th. List > obsss | Structured version Visualization version GIF version |
Description: An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsss.v | ⊢ 𝑉 = (Base‘𝑊) |
Ref | Expression |
---|---|
obsss | ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | obsss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2798 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2798 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2798 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
5 | eqid 2798 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | eqid 2798 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
7 | eqid 2798 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 20409 | . 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(·𝑖‘𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)}))) |
9 | 8 | simp2bi 1143 | 1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ifcif 4425 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑖cip 16562 0gc0g 16705 1rcur 19244 PreHilcphl 20313 ocvcocv 20349 OBasiscobs 20391 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-obs 20394 |
This theorem is referenced by: obsne0 20414 obselocv 20417 obslbs 20419 |
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