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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 2610 | . . . 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 2610 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 1, 2, 3, 4, 5 | obsip 19884 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
7 | 6 | 3anidm23 1377 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
8 | eqid 2610 | . . 3 ⊢ 𝐴 = 𝐴 | |
9 | 8 | iftruei 4043 | . 2 ⊢ if(𝐴 = 𝐴, 1 , (0g‘𝐹)) = 1 |
10 | 7, 9 | syl6eq 2660 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ifcif 4036 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑖cip 15773 0gc0g 15923 1rcur 18324 OBasiscobs 19865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-obs 19868 |
This theorem is referenced by: obsne0 19888 |
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