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Mirrors > Home > MPE Home > Th. List > obsipid | Structured version Visualization version GIF version |
Description: A basis element has length one. (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 2735 | . . . 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 2735 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 1, 2, 3, 4, 5 | obsip 21759 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
7 | 6 | 3anidm23 1420 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
8 | eqid 2735 | . . 3 ⊢ 𝐴 = 𝐴 | |
9 | 8 | iftruei 4538 | . 2 ⊢ if(𝐴 = 𝐴, 1 , (0g‘𝐹)) = 1 |
10 | 7, 9 | eqtrdi 2791 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Scalarcsca 17301 ·𝑖cip 17303 0gc0g 17486 1rcur 20199 OBasiscobs 21740 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-obs 21743 |
This theorem is referenced by: obsne0 21763 |
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