![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2731 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2731 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2731 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
5 | eqid 2731 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | eqid 2731 | . . 3 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
7 | eqid 2731 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 21585 | . 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 1144 | 1 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3948 ifcif 4528 {csn 4628 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 Scalarcsca 17207 ·𝑖cip 17209 0gc0g 17392 1rcur 20082 PreHilcphl 21487 ocvcocv 21523 OBasiscobs 21567 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-obs 21570 |
This theorem is referenced by: obsne0 21590 obs2ocv 21592 obselocv 21593 obslbs 21595 |
Copyright terms: Public domain | W3C validator |