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Mirrors > Home > MPE Home > Th. List > obsipid | Structured version Visualization version GIF version |
Description: A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsipid.h | ⊢ , = (·𝑖‘𝑊) |
obsipid.f | ⊢ 𝐹 = (Scalar‘𝑊) |
obsipid.u | ⊢ 1 = (1r‘𝐹) |
Ref | Expression |
---|---|
obsipid | ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | obsipid.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
3 | obsipid.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | obsipid.u | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | eqid 2821 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 1, 2, 3, 4, 5 | obsip 20859 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
7 | 6 | 3anidm23 1417 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
8 | eqid 2821 | . . 3 ⊢ 𝐴 = 𝐴 | |
9 | 8 | iftruei 4473 | . 2 ⊢ if(𝐴 = 𝐴, 1 , (0g‘𝐹)) = 1 |
10 | 7, 9 | syl6eq 2872 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ifcif 4466 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑖cip 16564 0gc0g 16707 1rcur 19245 OBasiscobs 20840 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-obs 20843 |
This theorem is referenced by: obsne0 20863 |
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