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Mirrors > Home > MPE Home > Th. List > obsocv | Structured version Visualization version GIF version |
Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsocv.z | ⊢ 0 = (0g‘𝑊) |
obsocv.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
obsocv | ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘𝐵) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2821 | . . . 4 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
5 | eqid 2821 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | obsocv.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
7 | obsocv.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 20847 | . . 3 ⊢ (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))) ∧ ( ⊥ ‘𝐵) = { 0 }))) |
9 | 8 | simp3bi 1143 | . 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(·𝑖‘𝑊)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘𝑊)), (0g‘(Scalar‘𝑊))) ∧ ( ⊥ ‘𝐵) = { 0 })) |
10 | 9 | simprd 498 | 1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘𝐵) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3924 ifcif 4453 {csn 4553 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 Scalarcsca 16551 ·𝑖cip 16553 0gc0g 16696 1rcur 19234 PreHilcphl 20751 ocvcocv 20787 OBasiscobs 20829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fv 6349 df-ov 7145 df-obs 20832 |
This theorem is referenced by: obs2ocv 20854 obs2ss 20856 |
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