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Mirrors > Home > MPE Home > Th. List > obsip | Structured version Visualization version GIF version |
Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
isobs.v | ⊢ 𝑉 = (Base‘𝑊) |
isobs.h | ⊢ , = (·𝑖‘𝑊) |
isobs.f | ⊢ 𝐹 = (Scalar‘𝑊) |
isobs.u | ⊢ 1 = (1r‘𝐹) |
isobs.z | ⊢ 0 = (0g‘𝐹) |
Ref | Expression |
---|---|
obsip | ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isobs.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
2 | isobs.h | . . . . . 6 ⊢ , = (·𝑖‘𝑊) | |
3 | isobs.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | isobs.u | . . . . . 6 ⊢ 1 = (1r‘𝐹) | |
5 | isobs.z | . . . . . 6 ⊢ 0 = (0g‘𝐹) | |
6 | eqid 2821 | . . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
7 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isobs 20863 | . . . . 5 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)}))) |
9 | 8 | simp3bi 1143 | . . . 4 ⊢ (𝐵 ∈ (OBasis‘𝑊) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ((ocv‘𝑊)‘𝐵) = {(0g‘𝑊)})) |
10 | 9 | simpld 497 | . . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )) |
11 | oveq1 7162 | . . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 , 𝑦) = (𝑃 , 𝑦)) | |
12 | eqeq1 2825 | . . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 = 𝑦 ↔ 𝑃 = 𝑦)) | |
13 | 12 | ifbid 4488 | . . . . 5 ⊢ (𝑥 = 𝑃 → if(𝑥 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑦, 1 , 0 )) |
14 | 11, 13 | eqeq12d 2837 | . . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ))) |
15 | oveq2 7163 | . . . . 5 ⊢ (𝑦 = 𝑄 → (𝑃 , 𝑦) = (𝑃 , 𝑄)) | |
16 | eqeq2 2833 | . . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 = 𝑦 ↔ 𝑃 = 𝑄)) | |
17 | 16 | ifbid 4488 | . . . . 5 ⊢ (𝑦 = 𝑄 → if(𝑃 = 𝑦, 1 , 0 ) = if(𝑃 = 𝑄, 1 , 0 )) |
18 | 15, 17 | eqeq12d 2837 | . . . 4 ⊢ (𝑦 = 𝑄 → ((𝑃 , 𝑦) = if(𝑃 = 𝑦, 1 , 0 ) ↔ (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))) |
19 | 14, 18 | rspc2v 3632 | . . 3 ⊢ ((𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))) |
20 | 10, 19 | syl5com 31 | . 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) → ((𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))) |
21 | 20 | 3impib 1112 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ifcif 4466 {csn 4566 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 Scalarcsca 16567 ·𝑖cip 16569 0gc0g 16712 1rcur 19250 PreHilcphl 20767 ocvcocv 20803 OBasiscobs 20845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-obs 20848 |
This theorem is referenced by: obsipid 20865 obselocv 20871 |
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