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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 2610 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2610 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2610 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
5 | eqid 2610 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | eqid 2610 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
7 | eqid 2610 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 19883 | . 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 1070 | 1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ifcif 4036 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑖cip 15773 0gc0g 15923 1rcur 18324 PreHilcphl 19788 ocvcocv 19823 OBasiscobs 19865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-obs 19868 |
This theorem is referenced by: obsne0 19888 obselocv 19891 obslbs 19893 |
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